Function Theory and Transformations

Function Definition and Mapping

In mathematics, a relation is any set of ordered pairs, such as \\((x,y)\\). A function is a special kind of relation.

A relation is a function if each input is matched with exactly one output. The input is usually called the independent variable or the domain value, and the output is called the dependent variable or the range value.

This lesson will help you decide whether a relation is a function by using:

• the definition of a function,
• mapping diagrams, and
• the vertical line test.

These ideas are important because before graphing, composing, or inverting functions, you must first know whether the relation is actually a function.

1. The definition of a function

Suppose a relation contains ordered pairs like \\((1,4)\\), \\((2,7)\\), and \\((3,9)\\). To decide whether it is a function, look only at the first coordinate, the input.

If one input appears with two different outputs, then the relation is not a function.

For example, the relation

$$\{(1,4),(2,7),(3,9)\}$$

is a function because each input has one output.

But the relation

$$\{(1,4),(2,7),(2,8),(3,9)\}$$

is not a function because the input \\(2\\) is matched with both \\(7\\) and \\(8\\).

Notice something important: different inputs can share the same output. That is still a function.

For example,

$$\{(1,5),(2,5),(3,5)\}$$

is a function because each input still has exactly one output.

2. Domain and range in mapping

The domain is the set of all inputs. The range is the set of outputs that actually occur.

In a mapping diagram, domain values are usually written in one oval on the left, and range values are written in another oval on the right. Arrows show which input maps to which output.

For a relation to be a function in a mapping diagram:

• each domain value must have one arrow leaving it,
• no domain value can have two different arrows going to different outputs,
• it is allowed for different inputs to point to the same output.

So when you read a mapping diagram, focus on the arrows coming out of the inputs.

3. How mapping diagrams show functions

Imagine the domain is \\({1,2,3}\\) and the outputs are \\({4,5}\\). If the arrows are

$$1 \to 4, \quad 2 \to 5, \quad 3 \to 5$$

then the relation is a function. Each input has exactly one output.

But if the arrows are

$$1 \to 4, \quad 2 \to 5, \quad 2 \to 4, \quad 3 \to 5$$

then it is not a function, because input \\(2\\) has two outputs.

4. Graphs and the vertical line test

When a relation is shown on a graph, we use the vertical line test to decide whether it is a function.

The rule is simple:

• If any vertical line crosses the graph at more than one point, the relation is not a function.
• If every vertical line crosses the graph at at most one point, the relation is a function.

Why does this work? A vertical line has one fixed \\(x\\)-value. If it crosses the graph more than once, that means the same input \\(x\\) gives more than one output \\(y\\). That breaks the definition of a function.

5. Examples of graphs that are functions

The graph of a line like

$$y=2x+3$$

is a function. Every \\(x\\)-value gives exactly one \\(y\\)-value, so any vertical line meets the graph only once.

The graph of a parabola like

$$y=x^2$$

is also a function. Even though some horizontal lines cross it twice, vertical lines still cross it only once. So it passes the vertical line test.

6. Examples of graphs that are not functions

The relation

$$x=y^2$$

is not a function of \\(x\\). Its graph is a sideways parabola. For many values of \\(x\\), there are two possible \\(y\\)-values, one positive and one negative.

For example, if \\(x=4\\), then

$$4=y^2$$

so \\(y=2\\) or \\(y=-2\\). One input has two outputs, so it is not a function.

A circle also fails the vertical line test. For example,

$$x^2+y^2=9$$

is not a function because many vertical lines hit the circle twice.

7. Common mistakes to avoid

• Mistake 1: Thinking repeated outputs mean it is not a function. Repeated outputs are allowed. Repeated inputs with different outputs are not allowed.
• Mistake 2: Looking at arrows going into an output in a mapping diagram. The key is how many arrows leave each input.
• Mistake 3: Using the horizontal line test instead of the vertical line test. The horizontal line test is for checking whether a function is one-to-one, not for checking whether it is a function at all.
• Mistake 4: Assuming every equation is a function. Some equations represent relations that are not functions.

Worked Example 1: Ordered pairs

Determine whether the relation is a function:

$$\{(-1,2),(0,5),(1,2),(3,7)\}$$

Step 1: Look at the inputs: \\(-1,0,1,3\\).

Step 2: Check whether any input repeats with a different output.

No input repeats.

Conclusion: This relation is a function.

Notice that the output \\(2\\) appears twice, but that is allowed because it comes from two different inputs.

Worked Example 2: Mapping diagram

A mapping diagram shows:

$$2 \to 6, \quad 4 \to 8, \quad 6 \to 8, \quad 8 \to 10$$

Is this a function?

Step 1: Check each input in the domain: \\(2,4,6,8\\).

Step 2: Count the arrows leaving each input.

• \\(2\\) has one arrow to \\(6\\)
• \\(4\\) has one arrow to \\(8\\)
• \\(6\\) has one arrow to \\(8\\)
• \\(8\\) has one arrow to \\(10\\)

Each input has exactly one output.

Conclusion: The relation is a function.

Worked Example 3: Mapping diagram that is not a function

A mapping diagram shows:

$$1 \to 3, \quad 2 \to 4, \quad 2 \to 5, \quad 3 \to 6$$

Determine whether it is a function.

Step 1: Look at input \\(2\\).

Step 2: Input \\(2\\) points to both \\(4\\) and \\(5\\).

That means one input has two outputs.

Conclusion: The relation is not a function.

Worked Example 4: Vertical line test

Determine whether the graph of

$$x=(y-1)^2$$

represents a function of \\(x\\).

Step 1: Recognize the shape. This is a sideways parabola opening to the right.

Step 2: Think about a vertical line, such as \\(x=4\\).

Substitute into the equation:

$$4=(y-1)^2$$

So

$$y-1=2 \quad \text{or} \quad y-1=-2$$

which gives

$$y=3 \quad \text{or} \quad y=-1$$

The same \\(x\\)-value gives two different \\(y\\)-values.

Conclusion: The graph does not represent a function.

8. Quick checklist for deciding if a relation is a function

1. If you have ordered pairs, check whether any input is paired with more than one output.
2. If you have a mapping diagram, check whether any input has more than one arrow leaving it.
3. If you have a graph, use the vertical line test.
4. Remember: many-to-one is allowed, but one-to-many is not.

9. Why this matters in function theory

In later work, you will study inverse functions, compositions, and transformations. All of those ideas depend on knowing whether a relation is a function first.

For example, if a relation is not a function, then it cannot behave like a normal input-output rule. That makes it harder or impossible to compose it with other functions in the usual way.

So identifying functions correctly is a basic but very important skill.

Summary

A function is a relation where each input has exactly one output. In ordered pairs, this means no input can appear with two different outputs. In mapping diagrams, each input must have exactly one arrow leaving it. On graphs, a relation is a function if it passes the vertical line test, meaning no vertical line crosses the graph more than once.

Domain and Range Analysis is about answering two important questions for a relation or function:

• Domain: What input values are allowed?
• Range: What output values can actually occur?

These ideas are central in function theory because they tell us where a formula makes sense and what values it can produce. They are also very important when working with inverse functions, transformations, and real-world models.

In Grade 12 mathematics, you may need to find domain and range from an equation, a graph, a table, or a word problem. You may also need to restrict a domain so that a function becomes invertible.

This lesson will show you how to analyze domain and range step by step.

1. What is the domain?

The domain is the set of all input values, usually the possible values of \(x\), for which the relation or function is defined.

When finding the domain from a formula, start by assuming all real numbers are allowed. Then remove any values that make the expression impossible or undefined.

In Grade 12, the most common restrictions come from:

• Division by zero is not allowed.
• Even roots, such as square roots, require the expression inside to be non-negative.
• In an applied context, inputs may be limited by the situation, such as time not being negative.

2. What is the range?

The range is the set of all output values, usually the possible values of \(y\), that the function can produce.

To find the range, you can:

• Study the shape of the graph.
• Use algebra to rewrite the function in a useful form.
• Consider restrictions from the domain.
• Think about minimum and maximum values.

The range depends on the function and the domain. If the domain changes, the range may also change.

3. Interval notation reminders

Domain and range are often written using interval notation.

• \((a,b)\) means all values between \(a\) and \(b\), not including the endpoints.
• \([a,b]\) means all values between \(a\) and \(b\), including both endpoints.
• \(( -\infty, a )\) or \((a, \infty)\) are always written with round brackets at infinity.

For example:

• \(x \neq 3\) can be written as $$(-\infty,3)\cup(3,\infty).$$
• \(x \ge 2\) can be written as $$[2,\infty).$$

4. How to find domain from algebraic formulas

Here is a useful process:

1. Start with all real numbers.
2. Check whether any denominator could become zero.
3. Check whether any square root or other even root has a negative value inside.
4. If the problem is from a real-world situation, apply practical restrictions.

5. How to find range from algebraic formulas

To find the range, ask: what values can \(y\) take?

Some helpful strategies are:

• Sketch or imagine the graph.
• Solve the equation \(y=f(x)\) for \(x\), then determine what values of \(y\) make sense.
• Rewrite the function to see its lowest or highest point.
• Use known shapes such as lines, parabolas, square root graphs, and reciprocal graphs.

6. Common graph behaviors and their domain/range

• Linear functions like \(y=2x-1\): domain is all real numbers, range is all real numbers.
• Quadratic functions like \(y=(x-3)^2+2\): domain is all real numbers, range depends on the vertex.
• Square root functions like \(y=\sqrt{x+1}\): domain comes from the radicand being non-negative, range is usually \(y\ge 0\).
• Reciprocal functions like \(y=\frac{1}{x-2}\): domain excludes where the denominator is zero, and range excludes values the function can never equal.

7. Worked Example 1: Rational function

Find the domain and range of

$$f(x)=\frac{3}{x-2}.$$

Step 1: Find the domain.

The denominator cannot be zero.

So,

$$x-2\neq 0$$

which gives

$$x\neq 2.$$

Therefore the domain is

$$(-\infty,2)\cup(2,\infty).$$

Step 2: Find the range.

The output is

$$y=\frac{3}{x-2}.$$

A fraction of this form can never equal zero, because the numerator is \(3\), not \(0\).

So,

$$y\neq 0.$$

Therefore the range is

$$(-\infty,0)\cup(0,\infty).$$

Answer:

• Domain: $$(-\infty,2)\cup(2,\infty)$$
• Range: $$(-\infty,0)\cup(0,\infty)$$

8. Worked Example 2: Square root function

Find the domain and range of

$$g(x)=\sqrt{5-x}.$$

Step 1: Find the domain.

The expression inside the square root must be non-negative:

$$5-x\ge 0.$$

Solving gives

$$x\le 5.$$

So the domain is

$$(-\infty,5].$$

Step 2: Find the range.

A square root output is always non-negative, so

$$g(x)\ge 0.$$

Can it be \(0\)? Yes. When \(x=5\),

$$g(5)=\sqrt{0}=0.$$

Can it become very large? Yes. If \(x\) becomes very negative, then \(5-x\) becomes very large, so the square root also becomes very large.

Therefore the range is

$$[0,\infty).$$

Answer:

• Domain: $$(-\infty,5]$$
• Range: $$[0,\infty)$$

9. Worked Example 3: Quadratic function and restricting domain for invertibility

Consider

$$h(x)=x^2-4x+1.$$

First find the domain and range. Then choose a restricted domain that makes the function invertible.

Step 1: Domain.

This is a quadratic polynomial, so it is defined for all real numbers.

Domain:

$$(-\infty,\infty).$$

Step 2: Rewrite to find the minimum value.

Complete the square:

$$h(x)=x^2-4x+1=(x-2)^2-3.$$

This shows the vertex is at \((2,-3)\).

Since \((x-2)^2\ge 0\), the smallest value of \(h(x)\) is \(-3\).

So the range is

$$[-3,\infty).$$

Step 3: Restrict the domain for invertibility.

A quadratic is not one-to-one on all real numbers, because many outputs come from two different inputs. For example, \(h(1)=h(3)=-2\).

To make it invertible, we restrict to one side of the vertex.

Two common choices are:

• \(x\ge 2\), giving domain $$[2,\infty),$$ or
• \(x\le 2\), giving domain $$(-\infty,2].$$

Either restriction makes the function one-to-one.

Important idea: Restricting the domain does not change the formula, but it changes which inputs are allowed. This is often done so an inverse function can exist.

10. Worked Example 4: Applied context

A ball is thrown upward, and its height after \(t\) seconds is modeled by

$$H(t)=-5t^2+20t+2.$$

Find a reasonable domain and range for the situation.

Step 1: Context-based domain.

Since \(t\) is time, we must have

$$t\ge 0.$$

But the model only makes sense until the ball hits the ground, when \(H(t)=0\).

So solve

$$-5t^2+20t+2=0.$$

Divide by \(-1\):

$$5t^2-20t-2=0.$$

Using the quadratic formula,

$$t=\frac{20\pm\sqrt{(-20)^2-4(5)(-2)}}{2(5)}$$

$$t=\frac{20\pm\sqrt{400+40}}{10}$$

$$t=\frac{20\pm\sqrt{440}}{10}$$

$$t=\frac{20\pm 2\sqrt{110}}{10}=2\pm\frac{\sqrt{110}}{5}.$$

The negative value is not physically meaningful, so the time ends at

$$t=2+\frac{\sqrt{110}}{5}.$$

Thus the practical domain is

$$\left[0,\,2+\frac{\sqrt{110}}{5}\right].$$

Step 2: Find the range.

The parabola opens downward, so its maximum is at the vertex.

For

$$H(t)=-5t^2+20t+2,$$

the vertex occurs at

$$t=\frac{-b}{2a}=\frac{-20}{2(-5)}=2.$$

Now find the height:

$$H(2)=-5(2)^2+20(2)+2=-20+40+2=22.$$

Since the ball starts above the ground and ends when it reaches the ground, the lowest height is \(0\) and the highest height is \(22\).

So the range is

$$[0,22].$$

Answer:

• Domain: $$\left[0,\,2+\frac{\sqrt{110}}{5}\right]$$
• Range: $$[0,22]$$

11. Finding domain and range from a graph

When working from a graph instead of a formula:

• The domain is found by looking left to right. Ask: for which \(x\)-values does the graph exist?
• The range is found by looking bottom to top. Ask: which \(y\)-values are reached by the graph?

Watch carefully for:

• Open circles: the endpoint is not included.
• Closed circles: the endpoint is included.
• Arrows: the graph continues forever.
• Breaks or gaps: values may be excluded.

12. Common mistakes to avoid

• Forgetting to exclude values that make a denominator zero.
• Allowing negative values inside a square root.
• Confusing domain and range.
• Giving the range as all real numbers without checking for a minimum or maximum.
• Ignoring the real-world context in applied problems.
• For quadratics, forgetting that restricting the domain can make the function invertible.

13. Quick checklist

When you are asked for domain and range, ask yourself:

1. What values of \(x\) are allowed?
2. Does anything make the formula undefined?
3. What outputs can the function actually produce?
4. Is there a minimum or maximum value?
5. Is there a graph or context that changes the answer?
6. Do I need to restrict the domain to make the function one-to-one?

14. Brief summary

The domain is the set of valid inputs, and the range is the set of possible outputs. To find the domain, remove values that make the formula undefined or impossible. To find the range, study the graph, the algebraic form, and any restrictions from the domain or context.

In many Grade 12 problems, domain and range analysis helps you understand function behavior, identify restrictions, and decide when a function can have an inverse. The key is to work carefully and check both the mathematics and the meaning of the situation.

Rates of Change and Difference Quotients

In many maths problems, we want to describe how fast one quantity changes compared to another. For example, how fast distance changes with time, how fast profit changes when sales increase, or how quickly a graph rises or falls.

This idea is called a rate of change. In function theory, rates of change help us understand the behavior of a function over an interval. They also lead directly to one of the most important ideas in higher maths: the derivative.

Before studying derivatives, students learn the difference quotient. The difference quotient measures how much a function changes between two inputs and compares that change to the change in the input values.

In this lesson, you will learn how to:

• find the average rate of change of a function over an interval,
• interpret rate of change from tables, graphs, and formulas,
• write and simplify the difference quotient,
• connect these ideas to the slope of a secant line.

1. Average Rate of Change

If a function changes from one input to another, the average rate of change tells us the change in output divided by the change in input.

For a function \(f(x)\) on the interval from \(x=a\) to \(x=b\), the average rate of change is

This formula compares:

• change in output: \(f(b)-f(a)\)
• change in input: \(b-a\)

This is the same idea as slope:

So the average rate of change of a function over an interval is the slope of the secant line joining the points \((a,f(a))\) and \((b,f(b))\).

2. Interpreting the Meaning

The sign of the rate of change tells us about the function's behavior:

• If the rate of change is positive, the function increases on that interval.
• If the rate of change is negative, the function decreases on that interval.
• If the rate of change is zero, the output does not change over that interval.

The size of the number matters too:

• A larger positive value means the function increases more quickly.
• A larger negative value in magnitude means the function decreases more quickly.

Units are also important. If \(x\) is measured in seconds and \(f(x)\) is measured in metres, then the rate of change is in metres per second.

3. Average Rate of Change from Different Representations

You may be asked to find average rate of change from:

• a formula,
• a table of values,
• a graph.

No matter how the function is given, the process is the same:

1. Identify the two input values.
2. Find the corresponding outputs.
3. Substitute into \(\dfrac{f(b)-f(a)}{b-a}\).
4. Simplify carefully.

4. The Difference Quotient

The difference quotient is a special algebraic expression that measures the average rate of change of a function over a very general interval.

Instead of using two specific inputs \(a\) and \(b\), we use:

• one input: \(x\)
• a second nearby input: \(x+h\)

Then the difference quotient is

Here, \(h\) represents the change in the input. Since the second input is \(x+h\), the input change is

So the numerator is the change in output, and the denominator is the change in input.

This expression is important because it gives the average rate of change between \(x\) and \(x+h\). Later, in calculus, we study what happens when \(h\) becomes very small.

5. Why Simplifying the Difference Quotient Matters

When simplifying \(\dfrac{f(x+h)-f(x)}{h}\), you should:

• substitute carefully for \(f(x+h)\),
• use brackets correctly,
• expand expressions when needed,
• combine like terms,
• factor if possible,
• cancel \(h\) only if it is a common factor of the entire numerator.

A very common mistake is forgetting brackets when subtracting \(f(x)\). Remember:

means the entire expression for \(f(x)\) must be subtracted.

Worked Example 1: Average Rate of Change from a Formula

Find the average rate of change of \(f(x)=3x^2-2x+1\) on the interval \([1,4]\).

Step 1: Find the outputs.

Step 2: Use the formula.

Answer: The average rate of change is \(13\).

Interpretation: Over the interval from \(x=1\) to \(x=4\), the function increases by an average of 13 units in output for each increase of 1 unit in input.

Worked Example 2: Average Rate of Change from a Table

A function is given by the table:

• \(x=2\), \(f(x)=7\)
• \(x=5\), \(f(x)=19\)

Find the average rate of change from \(x=2\) to \(x=5\).

Use the formula:

Answer: The average rate of change is \(4\).

This means that, on average, the output increases by 4 for every increase of 1 in the input over this interval.

Worked Example 3: Simplifying a Difference Quotient

Simplify the difference quotient for \(f(x)=x^2+5x\).

Start with

Step 1: Find \(f(x+h)\).

Expand:

Step 2: Subtract \(f(x)\).

Distribute the subtraction:

Combine like terms:

Step 3: Divide by \(h\).

Factor out \(h\):

Cancel the common factor:

Answer: The simplified difference quotient is

Worked Example 4: Difference Quotient with a Fraction

Simplify the difference quotient for \(f(x)=\dfrac{1}{x}\).

Start with

Step 1: Combine the fractions in the numerator.

Simplify the numerator:

So we get

Step 2: Divide by \(h\).

This is the same as multiplying by \(\dfrac{1}{h}\):

Answer: The simplified difference quotient is

6. Connection to Secant Lines

On a graph, the average rate of change between two points is the slope of the line passing through those points. This line is called a secant line.

If the points are \((a,f(a))\) and \((b,f(b))\), then the slope of the secant line is

This gives a geometric meaning to average rate of change. It helps describe whether the graph is rising or falling over an interval.

7. Common Mistakes to Avoid

• Reversing the order: If you use \(f(b)-f(a)\), then the denominator must be \(b-a\). Keep the order consistent.
• Forgetting brackets: Write \(f(x+h)\) carefully, especially when squaring binomials.
• Incorrect expansion: Remember that \((x+h)^2=x^2+2xh+h^2\), not \(x^2+h^2\).
• Cancelling too early: You can only cancel \(h\) if it is a factor of the entire numerator.
• Ignoring domain restrictions: In expressions like \(\dfrac{1}{x}\), values that make the denominator zero are not allowed.

8. Quick Practice Ideas

To become confident, practise these types of questions:

• Find the average rate of change of a quadratic over a given interval.
• Use a table of values to calculate change over an interval.
• Simplify the difference quotient for linear, quadratic, and simple rational functions.
• Interpret whether a function is increasing or decreasing based on the sign of the rate of change.

9. Summary

The average rate of change of a function on \([a,b]\) is

It tells how much the output changes per unit change in the input, and it is the slope of the secant line between two points on the graph.

The difference quotient is

It is an algebraic version of average rate of change and is an important stepping stone toward derivatives. To simplify it successfully, substitute carefully, expand correctly, combine like terms, and factor before cancelling.

Monotonicity and Extrema are two important ideas used to describe how a function behaves.

Monotonicity tells us whether a function is increasing, decreasing, or constant on an interval.

Extrema are the highest and lowest values of a function. These include local maxima and minima as well as absolute maxima and minima.

Understanding these ideas helps us describe graphs, solve real-world problems, and decide when a function can be restricted to become invertible.

1. Increasing, Decreasing, and Constant Functions

A function is increasing on an interval if its output values go up as the input values increase.

A function is decreasing on an interval if its output values go down as the input values increase.

A function is constant on an interval if its output values stay the same as the input values increase.

For example:

• If for larger values of \(x\), the values of \(f(x)\) become larger, then \(f\) is increasing.

• If for larger values of \(x\), the values of \(f(x)\) become smaller, then \(f\) is decreasing.

• If \(f(x)\) does not change, then the function is constant.

We usually describe monotonicity using interval notation, such as \((-8, 2)\) or \([1, 5]\).

2. Extrema: Maximum and Minimum Values

An extremum is either a maximum or a minimum value of a function.

• A local maximum is a point where the function is higher than nearby points.

• A local minimum is a point where the function is lower than nearby points.

• An absolute maximum is the highest value of the function on its domain or a given interval.

• An absolute minimum is the lowest value of the function on its domain or a given interval.

A function can have:

• local extrema but no absolute extrema,

• absolute extrema at endpoints,

• more than one local extremum,

• or no extrema at all.

3. Reading Monotonicity from a Graph

When reading a graph from left to right:

• If the graph goes upward, the function is increasing.

• If the graph goes downward, the function is decreasing.

• If the graph is flat, the function is constant.

Turning points are especially important.

If the graph changes from increasing to decreasing, there is a local maximum.

If the graph changes from decreasing to increasing, there is a local minimum.

4. Using Derivatives to Find Monotonicity

In Grade 12, the derivative is a powerful tool for studying how a function behaves.

If \(f'(x) > 0\) on an interval, then \(f(x)\) is increasing on that interval.

If \(f'(x) < 0\) on an interval, then \(f(x)\) is decreasing on that interval.

If \(f'(x) = 0\) throughout an interval, then \(f(x)\) is constant on that interval.

So the sign of the derivative tells us the monotonicity of the function.

5. Critical Points

A critical point happens where:

• \(f'(x) = 0\), or

• \(f'(x)\) does not exist, but \(f(x)\) does exist.

Critical points are important because local maxima and minima often occur there.

However, not every critical point is an extremum. Sometimes the function continues increasing or continues decreasing.

6. First Derivative Test

To decide whether a critical point is a maximum or minimum, we examine the sign of \(f'(x)\) before and after that point.

• If \(f'(x)\) changes from positive to negative, the function changes from increasing to decreasing, so there is a local maximum.

• If \(f'(x)\) changes from negative to positive, the function changes from decreasing to increasing, so there is a local minimum.

• If the sign does not change, then there is no local extremum there.

7. Absolute Extrema on a Closed Interval

To find the absolute maximum and minimum of a continuous function on a closed interval \([a,b]\):

1. Find all critical points inside the interval.

2. Evaluate the function at each critical point.

3. Evaluate the function at the endpoints \(x=a\) and \(x=b\).

4. Compare all these values.

The largest value is the absolute maximum, and the smallest value is the absolute minimum.

8. Why Monotonicity Matters for Invertibility

A function must pass the horizontal line test to have an inverse function.

If a function is strictly increasing or strictly decreasing on an interval, then it is one-to-one on that interval, so it is invertible there.

This is why we often restrict the domain of a function. For example, the quadratic function \(f(x)=x^2\) is not one-to-one on all real numbers, but it becomes invertible if we restrict the domain to \(x \ge 0\) or \(x \le 0\).

On each of those restricted domains, the function is monotonic.

Worked Example 1: Monotonicity from a Simple Function

Determine where \(f(x)=2x+3\) is increasing or decreasing.

Step 1: Find the derivative.

Step 2: Analyze the sign of the derivative.

Since \(2 > 0\) for all \(x\), the derivative is always positive.

Conclusion: The function is increasing on

It is not decreasing on any interval, and it has no local or absolute maximum or minimum.

This makes sense because a straight line with positive slope always rises from left to right.

Worked Example 2: Finding Local Extrema

Find the intervals of increase and decrease for \(f(x)=x^2-4x+1\), and identify any local extrema.

Step 1: Differentiate.

Step 2: Find critical points.

Step 3: Test the sign of \(f'(x)\).

• For \(x<2\), choose \(x=0\):

  $$f'(0)=2(0)-4=-4<0$$

  So the function is decreasing on \(( -\infty, 2 )\).

• For \(x>2\), choose \(x=3\):

  $$f'(3)=2(3)-4=2>0$$

  So the function is increasing on \((2, \infty)\).

Step 4: Identify the extremum.

The derivative changes from negative to positive at \(x=2\), so there is a local minimum.

Now find the function value:

Conclusion:

• Decreasing on \(( -\infty, 2 )\)

• Increasing on \((2, \infty)\)

• Local minimum at \((2,-3)\)

Because this parabola opens upward, this local minimum is also the absolute minimum.

Worked Example 3: Local Maximum and Minimum

Analyze \(f(x)=x^3-3x^2+2\). Find intervals of increase and decrease, and identify local extrema.

Step 1: Differentiate.

Factor:

Step 2: Find critical points.

So the critical points are:

Step 3: Test intervals.

• For \(x<0\), choose \(x=-1\):

  $$f'(-1)=3(-1)(-3)=9>0$$

  Increasing on \(( -\infty, 0 )\)

• For \(0<x<2\), choose \(x=1\):

  $$f'(1)=3(1)(-1)=-3<0$$

  Decreasing on \((0,2)\)

• For \(x>2\), choose \(x=3\):

  $$f'(3)=3(3)(1)=9>0$$

  Increasing on \((2,\infty)\)

Step 4: Classify the extrema.

At \(x=0\), the derivative changes from positive to negative, so there is a local maximum.

At \(x=2\), the derivative changes from negative to positive, so there is a local minimum.

Now find the function values:

Conclusion:

• Increasing on \(( -\infty,0 )\) and \((2,\infty)\)

• Decreasing on \((0,2)\)

• Local maximum at \((0,2)\)

• Local minimum at \((2,-2)\)

This cubic function has both a local maximum and a local minimum, but it has no absolute maximum or minimum over all real numbers because it continues forever upward and downward.

Worked Example 4: Absolute Extrema on a Closed Interval

Find the absolute maximum and minimum of \(f(x)=x^3-3x\) on the interval \([-2,2]\).

Step 1: Differentiate.

Step 2: Find critical points in the interval.

Both \(-1\) and \(1\) lie in \([-2,2]\).

Step 3: Evaluate the function at critical points and endpoints.

Step 4: Compare values.

• Largest value: \(2\)

• Smallest value: \(-2\)

Conclusion:

• Absolute maximum value is \(2\), occurring at \(x=-1\) and \(x=2\)

• Absolute minimum value is \(-2\), occurring at \(x=-2\) and \(x=1\)

This example shows that an absolute maximum or minimum can occur at more than one point.

9. Common Mistakes to Avoid

• Confusing local and absolute extrema: A local maximum is only higher than nearby points, not necessarily the highest point on the entire domain.

• Forgetting endpoints: On a closed interval, absolute extrema may occur at endpoints.

• Stopping after finding critical points: You must still test the sign of the derivative or compare function values.

• Assuming every critical point is an extremum: Some critical points are neither maxima nor minima.

• Ignoring the domain: Extrema and monotonicity always depend on the interval or domain being considered.

10. Step-by-Step Strategy

When asked to analyze monotonicity and extrema:

1. Find the derivative \(f'(x)\).

2. Solve \(f'(x)=0\) and note where \(f'(x)\) is undefined.

3. Use these points to divide the number line into intervals.

4. Test the sign of \(f'(x)\) in each interval.

5. State where the function is increasing or decreasing.

6. Use sign changes to identify local maxima and minima.

7. If working on a closed interval, also check endpoints for absolute extrema.

Brief Summary

Monotonicity describes whether a function increases, decreases, or stays constant on an interval.

Extrema are the highest and lowest values of a function, either locally or absolutely.

Derivatives help us find critical points and determine where a function is increasing or decreasing.

By checking the sign of \(f'(x)\) and comparing function values, we can locate local and absolute maxima and minima with confidence.

Symmetry and Parity help us describe how a function behaves when the input changes from \(x\) to \(-x\). This is an important idea in function theory because it connects algebra, graphs, and patterns in equations.

When studying a function, one useful question is: What happens if we replace \(x\) with \(-x\)? The answer lets us classify the function as even, odd, or neither.

These classifications are closely connected to graph symmetry. An even function has symmetry across the y-axis. An odd function has symmetry about the origin. If neither type of symmetry appears, the function is neither even nor odd.

1. What is an even function?

A function \(f(x)\) is even if replacing \(x\) with \(-x\) gives the same output:

This means the left side of the graph is a mirror image of the right side across the y-axis.

Common examples of even functions include:

• \(f(x)=x^2\)
• \(f(x)=x^4+3\)
• \(f(x)=|x|\)

For example, if \(f(x)=x^2\), then

So \(x^2\) is even.

2. What is an odd function?

A function \(f(x)\) is odd if replacing \(x\) with \(-x\) gives the opposite of the original output:

This means the graph has symmetry about the origin. If you rotate the graph \(180^\circ\) around the origin, it matches itself.

Common examples of odd functions include:

• \(f(x)=x^3\)
• \(f(x)=x\)
• \(f(x)=x^5-2x\)

For example, if \(f(x)=x^3\), then

So \(x^3\) is odd.

3. What does neither mean?

If a function does not satisfy either condition, then it is neither even nor odd.

For example, let \(f(x)=x^2+x\). Then

This is not equal to \(f(x)=x^2+x\), and it is also not equal to

So the function is neither even nor odd.

4. How to test parity algebraically

To classify a function, follow these steps:

1. Start with the original function \(f(x)\).
2. Replace every \(x\) with \(-x\) to find \(f(-x)\).
3. Compare \(f(-x)\) with \(f(x)\) and with \(-f(x)\).
4. Decide:

• If \(f(-x)=f(x)\), the function is even.
• If \(f(-x)=-f(x)\), the function is odd.
• If neither is true, the function is neither.

Be careful with signs, especially when substituting into powers, fractions, or radicals.

5. Graphical meaning of parity

Parity can also be seen from a graph.

• Even function: symmetric about the y-axis.
• Odd function: symmetric about the origin.
• Neither: no such symmetry.

If the point \((a,b)\) is on an even function, then \((-a,b)\) must also be on the graph.

If the point \((a,b)\) is on an odd function, then \((-a,-b)\) must also be on the graph.

6. Important note about the domain

To be even or odd, a function must have a domain that is symmetric about \(0\). That means whenever \(x\) is in the domain, \(-x\) must also be in the domain.

For example, a function defined only for \(x\ge 0\) usually cannot be classified as even or odd over that domain, because negative inputs are missing.

This is important in function theory, especially when we restrict domains for invertibility. A function may be even or odd on its full domain, but if we restrict the domain, that symmetry may no longer apply.

7. Parity of common types of terms

Polynomial functions are often easy to classify by looking at their powers.

• Terms with even powers, such as \(x^2\), \(x^4\), and constants, are even.
• Terms with odd powers, such as \(x\), \(x^3\), and \(x^5\), are odd.

From this, we can often decide:

• A sum of only even-power terms is usually even.
• A sum of only odd-power terms with no constant term is usually odd.
• A mix of even and odd terms is usually neither.

Examples:

• \(x^4-2x^2+7\) is even.
• \(3x^5-x\) is odd.
• \(x^3+x^2\) is neither.

8. Worked Example 1: Classify \(f(x)=x^2-5\)

Step 1: Find \(f(-x)\).

Step 2: Compare with \(f(x)\).

Since

the function is even.

Graph meaning: The graph is symmetric across the y-axis.

9. Worked Example 2: Classify \(g(x)=2x^3-6x\)

Step 1: Find \(g(-x)\).

Step 2: Find \(-g(x)\).

Since

the function is odd.

Graph meaning: The graph has symmetry about the origin.

10. Worked Example 3: Classify \(h(x)=x^3+x^2\)

Step 1: Find \(h(-x)\).

Step 2: Compare with \(h(x)\) and \(-h(x)\).

The expression \(-x^3+x^2\) is not equal to \(x^3+x^2\), and it is not equal to \(-x^3-x^2\).

So \(h(x)\) is neither even nor odd.

Reason: The function contains both an odd-power term and an even-power term.

11. Worked Example 4: Classify \(p(x)=\dfrac{x}{x^2+1}\)

This example shows that parity also applies to rational functions.

Step 1: Find \(p(-x)\).

Step 2: Compare with \(-p(x)\).

Since

the function is odd.

12. Quick checks and useful patterns

• If all exponents of \(x\) are even, think even.
• If all exponents of \(x\) are odd and there is no constant term, think odd.
• If the function mixes even and odd behavior, think neither.
• Always verify by substitution if you are unsure.

13. Common mistakes to avoid

• Forgetting parentheses: Write \(f(-x)\) carefully. For example, if \(f(x)=x^2-3x\), then

• Confusing y-axis symmetry with origin symmetry: Even means y-axis; odd means origin.
• Ignoring the domain: A function cannot be even or odd on a domain that does not include matching positive and negative inputs.
• Assuming every function must be one or the other: Many functions are neither.

14. Why symmetry and parity matter

Symmetry and parity help us understand a function more quickly. They can make graphing easier, simplify algebraic work, and reveal patterns in models.

In transformations, symmetry can change if a graph is shifted. For example, \(y=x^2\) is even, but \(y=(x-1)^2\) is not even because the graph has been moved to the right and is no longer symmetric about the y-axis.

So when analyzing a transformed function, do not assume it keeps the parity of the original one. Always test it again.

Brief Summary

A function is even if \(f(-x)=f(x)\), which means its graph is symmetric about the y-axis. A function is odd if \(f(-x)=-f(x)\), which means its graph is symmetric about the origin. If neither condition is true, the function is neither.

To classify a function, substitute \(-x\) for \(x\), simplify carefully, and compare. Also remember that the domain must be symmetric about \(0\) if a function is to be even or odd.

Piecewise-Defined and Step Functions are functions whose rules can change depending on the input value. Instead of using one formula for every number in the domain, the function is split into parts, and each part has its own rule.

These functions are important because many real-world situations do not follow one single pattern. For example, taxi fares may change after a certain distance, phone plans may charge different rates after a usage limit, and rounding functions behave differently on different intervals.

In this lesson, you will learn how to read, evaluate, and graph piecewise-defined functions, including two special step functions: the greatest integer (floor) function and the ceiling function.

1. What is a piecewise-defined function?

A piecewise-defined function is a function written using two or more formulas, where each formula applies to a certain part of the domain.

For example, consider

This means:

• Use the rule \(2x+1\) when \(x<0\).
• Use the rule \(x^2\) when \(x\ge 0\).

The key idea is that the input value tells you which rule to use.

2. How to evaluate a piecewise-defined function

To evaluate a piecewise function at a certain input:

1. Look at the input value.
2. Decide which interval or condition the input satisfies.
3. Use only the formula that matches that condition.
4. Substitute the value and simplify.

Worked Example 1: Evaluating a piecewise function

Let

Find \(g(-1)\), \(g(2)\), and \(g(5)\).

Step 1: Find \(g(-1)\)

Since \(-1<2\), use the rule \(x+4\).

Step 2: Find \(g(2)\)

Since \(2\ge 2\), use the rule \(3x-1\).

Step 3: Find \(g(5)\)

Since \(5\ge 2\), use the rule \(3x-1\).

So the answers are:

• \(g(-1)=3\)
• \(g(2)=5\)
• \(g(5)=14\)

3. How to graph a piecewise-defined function

To graph a piecewise function, graph each rule only on the interval where it is allowed.

Pay close attention to the endpoints of each interval:

• If the endpoint is included, use a closed dot.
• If the endpoint is not included, use an open dot.

For example, graph

For \(x<1\), graph the line \(y=x+1\), but only to the left of \(x=1\). At \(x=1\), this rule would give \(y=2\), but since \(x=1\) is not included, place an open dot at \((1,2)\).

For \(x\ge 1\), graph the horizontal line \(y=4\) starting at \(x=1\). Since \(x=1\) is included here, place a closed dot at \((1,4)\).

This graph has a jump at \(x=1\), because the function value changes suddenly from near \(2\) to \(4\).

4. Important ideas when reading piecewise functions

• Domain: The set of input values for which the function is defined.
• Range: The set of output values the function produces.
• Breakpoint: A value where the rule changes.
• Open circle: The point is not included.
• Closed circle: The point is included.

Many piecewise functions have behavior that changes at one or more breakpoints. These changes may create corners, jumps, or separate pieces on the graph.

5. Step functions

A step function is a piecewise-defined function whose graph looks like a staircase. It stays constant on intervals and then jumps to a new value.

One of the most important step functions is the greatest integer function, also called the floor function.

6. The floor function

The floor function is written as \(\lfloor x \rfloor\). It means the greatest integer less than or equal to \(x\).

Examples:

• \(\lfloor 3.7 \rfloor = 3\)
• \(\lfloor 5 \rfloor = 5\)
• \(\lfloor -1.2 \rfloor = -2\)
• \(\lfloor -4.9 \rfloor = -5\)

The negative examples are especially important. For \(-1.2\), the greatest integer less than or equal to \(-1.2\) is \(-2\), not \(-1\).

The floor function can be described piece by piece. For every integer \(n\),

This means the graph is made of horizontal segments. On each interval from \(n\) to just before \(n+1\), the value stays at \(n\).

Each step has:

• a closed dot at the left endpoint, because \(x=n\) is included,
• an open dot at the right endpoint, because \(x=n+1\) belongs to the next step instead.

Worked Example 2: Evaluating the floor function

Find the following:

• \(\lfloor 2.9 \rfloor\)
• \(\lfloor -0.1 \rfloor\)
• \(\lfloor 7 \rfloor\)

Solution

because 2 is the greatest integer less than or equal to 2.9.

because \(-1\) is the greatest integer less than or equal to \(-0.1\).

because 7 is already an integer.

7. The ceiling function

The ceiling function is written as \(\lceil x \rceil\). It means the smallest integer greater than or equal to \(x\).

Examples:

• \(\lceil 3.2 \rceil = 4\)
• \(\lceil 6 \rceil = 6\)
• \(\lceil -2.7 \rceil = -2\)
• \(\lceil -5.01 \rceil = -5\)

For every integer \(n\), the ceiling function satisfies

Its graph is also a staircase, but the endpoints are reversed compared with the floor function:

• an open dot at the left endpoint,
• a closed dot at the right endpoint.

Worked Example 3: Evaluating the ceiling function

Find the following:

• \(\lceil 4.01 \rceil\)
• \(\lceil -3.8 \rceil\)
• \(\lceil -2 \rceil\)

Solution

because 5 is the smallest integer greater than or equal to 4.01.

because \(-3\) is the smallest integer greater than or equal to \(-3.8\).

because \(-2\) is already an integer.

8. Comparing floor and ceiling

• The floor of a number rounds it down to the nearest integer at or below it.
• The ceiling of a number rounds it up to the nearest integer at or above it.

Be careful with negative numbers:

• \(\lfloor -2.3 \rfloor = -3\)
• \(\lceil -2.3 \rceil = -2\)

9. Graphing floor and ceiling functions

To graph \(y=\lfloor x \rfloor\):

• Draw horizontal segments on intervals \([n,n+1)\).
• Each segment has height \(n\).
• Use a closed dot at \((n,n)\) and an open dot at \((n+1,n)\).

To graph \(y=\lceil x \rceil\):

• Draw horizontal segments on intervals \((n-1,n]\).
• Each segment has height \(n\).
• Use an open dot at the left end and a closed dot at the right end.

Both graphs have jumps at every integer value.

10. Piecewise form of step functions

Step functions can often be written in piecewise form over a limited interval.

For example, on the interval \(-2 \le x < 2\), the floor function can be written as

This shows clearly why the graph is called a step function.

11. Worked Example 4: Evaluating and interpreting a more complex piecewise function

Let

Find:

• \(p(-2)\)
• \(p(0)\)
• \(p(3)\)

Then describe how to graph it.

Step 1: Evaluate \(p(-2)\)

Since \(-2<-1\), use \(x^2\).

Step 2: Evaluate \(p(0)\)

Since \(-1 \le 0 < 3\), use the constant rule \(2\).

Step 3: Evaluate \(p(3)\)

Since \(3\ge 3\), use \(-x+6\).

So:

• \(p(-2)=4\)
• \(p(0)=2\)
• \(p(3)=3\)

How to graph it:

• Graph \(y=x^2\) only for \(x<-1\). At \(x=-1\), the point \(( -1,1 )\) is not included, so use an open dot there.
• Graph the horizontal line \(y=2\) for \(-1 \le x < 3\). Use a closed dot at \((-1,2)\) and an open dot at \((3,2)\).
• Graph the line \(y=-x+6\) for \(x\ge 3\). Since \(x=3\) is included, use a closed dot at \((3,3)\).

This graph has different types of pieces: a curved part, a horizontal part, and a line segment extending to the right.

12. Common mistakes to avoid

• Using the wrong rule: Always check which interval the input belongs to before substituting.
• Ignoring endpoint symbols: Remember that \(<\) and \(>\) mean not included, while \(\le\) and \(\ge\) mean included.
• Confusing floor and ceiling: Floor is greatest integer less than or equal to \(x\); ceiling is smallest integer greater than or equal to \(x\).
• Mistakes with negatives: For example, \(\lfloor -1.4 \rfloor\) is \(-2\), not \(-1\).
• Graphing the full formula: Only graph each rule on its specified interval.

13. Why these functions matter

Piecewise-defined and step functions are useful in modeling situations where behavior changes at certain points. They can describe tax brackets, shipping costs, parking fees, phone billing, or any situation where rates stay constant for a range and then jump.

They also help build a stronger understanding of how functions can behave differently on different parts of their domains.

Brief Summary

A piecewise-defined function uses different formulas on different parts of its domain. To evaluate one, first identify which condition the input satisfies, then use the matching rule.

A step function has a graph made of horizontal pieces with jumps. The floor function \(\lfloor x \rfloor\) gives the greatest integer less than or equal to \(x\), and the ceiling function \(\lceil x \rceil\) gives the smallest integer greater than or equal to \(x\).

When graphing these functions, be careful with intervals and endpoints. Open and closed dots show whether values are included, and this is essential for drawing the graph correctly.

Geometric Transformations of Functions help us understand how the graph of a known function changes when its equation is changed. Instead of graphing every new function from scratch, we can begin with a parent function and describe how the new graph is shifted, reflected, stretched, or compressed.

This idea is very useful in Grade 12 Maths because many functions are related to each other by transformations. For example, the graph of \(y=(x-3)^2+2\) is just the graph of \(y=x^2\) moved to a new position. If you can recognize these changes, you can sketch graphs faster and understand how equations control shape and location.

In this lesson, we will connect algebraic changes in an equation to geometric changes on a graph. We will focus on translations, reflections, and dilations.

1. Parent functions

A parent function is the simplest form of a family of functions. Some common parent functions are:

• Linear: \(f(x)=x\)
• Quadratic: \(f(x)=x^2\)
• Cubic: \(f(x)=x^3\)
• Absolute value: \(f(x)=|x|\)
• Square root: \(f(x)=\sqrt{x}\)
• Reciprocal: \(f(x)=\frac{1}{x}\)

When a function is transformed, its graph changes position, orientation, or size, but it is still based on the same parent shape.

2. The general transformation form

A very useful form is:

Each part affects the graph in a specific way:

• \(h\): horizontal shift
• \(k\): vertical shift
• \(a\): vertical stretch/compression and possible reflection
• \(b\): horizontal stretch/compression and possible reflection

We will study each transformation one at a time.

3. Translations

A translation moves a graph without changing its shape or turning it.

Vertical translations

If we change \(f(x)\) to \(f(x)+k\), the graph moves:

• up by \(k\) units if \(k>0\)
• down by \(|k|\) units if \(k<0\)

Examples:

• \(y=x^2+4\): move \(y=x^2\) up 4 units
• \(y=|x|-2\): move \(y=|x|\) down 2 units

Horizontal translations

If we change \(f(x)\) to \(f(x-h)\), the graph moves:

• right by \(h\) units if \(h>0\)
• left by \(|h|\) units if \(h<0\)

This often feels backwards at first. For example, \(f(x-3)\) moves the graph right, not left.

Examples:

• \(y=(x-5)^2\): move \(y=x^2\) right 5 units
• \(y=\sqrt{x+2}\): move \(y=\sqrt{x}\) left 2 units

4. Reflections

A reflection flips a graph across an axis.

Reflection in the x-axis

If we change \(f(x)\) to \(-f(x)\), every \(y\)-value changes sign. The graph is reflected in the x-axis.

Examples:

• \(y=-x^2\): reflection of \(y=x^2\) in the x-axis
• \(y=-|x|\): reflection of \(y=|x|\) in the x-axis

Reflection in the y-axis

If we change \(f(x)\) to \(f(-x)\), every \(x\)-value changes sign. The graph is reflected in the y-axis.

Examples:

• \(y=(-x)^3=-x^3\): reflection of \(y=x^3\) in the y-axis
• \(y=|{-x}|=|x|\): the absolute value graph looks unchanged because it is already symmetric about the y-axis

5. Dilations: stretches and compressions

A dilation changes the size of a graph. It may stretch it or compress it.

Vertical dilation

If we change \(f(x)\) to \(a f(x)\):

• if \(|a|>1\), the graph is stretched vertically
• if \(0<|a|<1\), the graph is compressed vertically
• if \(a<0\), there is also a reflection in the x-axis

Examples:

• \(y=3x^2\): vertical stretch by factor 3
• \(y=\frac{1}{2}x^2\): vertical compression by factor \(\frac{1}{2}\)
• \(y=-2|x|\): reflection in x-axis and vertical stretch by factor 2

Horizontal dilation

If we change \(f(x)\) to \(f(bx)\):

• if \(|b|>1\), the graph is compressed horizontally by factor \(\frac{1}{|b|}\)
• if \(0<|b|<1\), the graph is stretched horizontally by factor \(\frac{1}{|b|}\)
• if \(b<0\), there is also a reflection in the y-axis

Examples:

• \(y=(2x)^2\): horizontal compression by factor \(\frac{1}{2}\)
• \(y=\sqrt{\frac{x}{3}}\): horizontal stretch by factor 3
• \(y=f(-2x)\): reflection in y-axis and horizontal compression by factor \(\frac{1}{2}\)

Important note: horizontal changes work differently from vertical changes. A number inside the function affects the graph in the horizontal direction, and the effect is the reciprocal of what many students first expect.

6. A transformation guide

• \(y=f(x)+k\): up/down
• \(y=f(x-h)\): left/right
• \(y=-f(x)\): reflect in x-axis
• \(y=f(-x)\): reflect in y-axis
• \(y=af(x)\): vertical stretch/compression
• \(y=f(bx)\): horizontal stretch/compression

7. Order of transformations

When a function has more than one transformation, it helps to read it carefully from the equation. In general:

1. Look at changes inside the function first: horizontal shift, horizontal dilation, reflection in y-axis.
2. Then look at changes outside the function: vertical dilation, reflection in x-axis, vertical shift.

For graphing, many students find it easiest to start with the parent graph and then apply the transformations step by step.

8. Worked Example 1: Simple translation

Describe the transformation from \(y=x^2\) to \(y=(x-3)^2+2\).

Step 1: Identify the parent function. It is \(f(x)=x^2\).

Step 2: Look at the inside change \((x-3)\). This moves the graph right 3.

Step 3: Look at the outside change \(+2\). This moves the graph up 2.

Result: The graph of \(y=x^2\) is translated 3 units right and 2 units up.

The vertex of the parent graph is \((0,0)\). After the transformation, the vertex becomes \((3,2)\).

9. Worked Example 2: Reflection and vertical stretch

Describe the transformation from \(y=|x|\) to \(y=-2|x|+1\).

Step 1: Parent function: \(f(x)=|x|\).

Step 2: The factor \(-2\) is outside the function.

• The negative sign reflects the graph in the x-axis.
• The factor 2 gives a vertical stretch by factor 2.

Step 3: The \(+1\) moves the graph up 1 unit.

Result: Reflect in the x-axis, stretch vertically by factor 2, then move up 1 unit.

The vertex moves from \((0,0)\) to \((0,1)\).

10. Worked Example 3: Horizontal compression and translation

Describe the transformation from \(y=\sqrt{x}\) to \(y=\sqrt{2(x-4)}\).

Step 1: Parent function: \(f(x)=\sqrt{x}\).

Step 2: The expression inside is \(2(x-4)\).

This tells us two horizontal transformations:

• \((x-4)\): move right 4 units
• the factor 2 inside: horizontal compression by factor \(\frac{1}{2}\)

Result: The graph of \(y=\sqrt{x}\) is moved right 4 units and compressed horizontally by factor \(\frac{1}{2}\).

To check, the parent graph starts at \((0,0)\). The transformed graph starts where the inside equals 0:

So the new starting point is \((4,0)\).

11. Worked Example 4: Full transformation

Analyze the function

relative to the parent graph \(y=f(x)\).

Step 1: Inside the function, we have \(-(x+2)\).

• \(x+2\) means shift left 2 units.
• The negative inside means reflect in the y-axis.

Step 2: Outside the function, we have \(-3\).

• The negative reflects in the x-axis.
• The factor 3 stretches vertically by factor 3.

Step 3: The \(+5\) shifts the graph up 5 units.

Result:

• shift left 2
• reflect in the y-axis
• reflect in the x-axis
• stretch vertically by factor 3
• shift up 5

12. How transformations affect key points

One of the fastest ways to sketch a transformed graph is to move key points from the parent graph.

For example, the parent parabola \(y=x^2\) has points:

• \((-2,4)\)
• \((-1,1)\)
• \((0,0)\)
• \((1,1)\)
• \((2,4)\)

For \(y=(x-1)^2-3\), the graph moves right 1 and down 3, so the points become:

• \((-1,1)\)
• \((0,-2)\)
• \((1,-3)\)
• \((2,-2)\)
• \((3,1)\)

This method helps you sketch accurately without making a table from scratch.

13. Common mistakes to avoid

• Horizontal shifts seem reversed. \(f(x-3)\) moves right, while \(f(x+3)\) moves left.
• Inside and outside changes are different. Outside affects vertical changes; inside affects horizontal changes.
• Horizontal scaling uses reciprocals. \(f(2x)\) compresses by \(\frac{1}{2}\), not by 2.
• Reflections depend on where the negative sign is. \(-f(x)\) reflects in the x-axis, but \(f(-x)\) reflects in the y-axis.

14. Why this matters

Geometric transformations allow you to compare functions, sketch them quickly, and understand how formulas model real situations. In later work, transformed functions are used to represent motion, growth, signals, and changing systems.

If you can identify the parent function and read the meaning of each algebraic change, then you can understand almost any transformed graph.

Brief Summary

Geometric transformations change a graph by moving it, flipping it, or resizing it. The main transformations are translations, reflections, and dilations. Changes outside the function affect the graph vertically, while changes inside the function affect it horizontally. By starting with a parent function and applying transformations step by step, you can sketch and analyze graphs with confidence.

Function Composition means putting one function inside another. It is like doing one process first, then taking that result and using it in a second process.

If a function is a rule that changes an input, then a composite function combines two rules into one chain. This is very useful when a situation happens in steps, such as converting units and then applying a cost rule, or finding an area and then using that area in another formula.

In symbols, the composition of functions \(f\) and \(g\) is written as \((f \circ g)(x)\). It is read as “\(f\) composed with \(g\)” or “\(f\) of \(g(x)\).”

The definition is:

This means:

• Start with \(x\).
• Apply \(g\) first.
• Then apply \(f\) to the result.

Important: Composition is usually not commutative. That means

in most cases. So the order matters.

For example, if \(f(x)=x+3\) and \(g(x)=2x\), then

but

These are different, so you must pay close attention to which function happens first.

How to compose functions

1. Identify the inside function.
2. Substitute the entire inside function into the outside function.
3. Simplify carefully.
4. Check the domain of the composite function.

The domain is especially important in composition. A value of \(x\) can only be used if:

• it is allowed in the inside function, and
• the output from the inside function is allowed as an input to the outside function.

So for \((f \circ g)(x)=f(g(x))\), the domain is all \(x\) such that:

• \(x\) is in the domain of \(g\), and
• \(g(x)\) is in the domain of \(f\).

This idea is very important when working with square roots, fractions, logarithms, and inverse functions.

Worked Example 1: Basic composition

Let

Find \((f \circ g)(x)\) and \((g \circ f)(x)\).

First, compose \(f\) with \(g\):

Now substitute \(x^2+4\) into \(f(x)=3x-1\):

Next, compose \(g\) with \(f\):

Now substitute \(3x-1\) into \(g(x)=x^2+4\):

So,

This example shows clearly that changing the order changes the answer.

Worked Example 2: Composition with a restricted domain

Let

Find \((f \circ g)(x)\) and its domain.

Start with the composition:

Substitute \(x+5\) into \(f(x)=\sqrt{x-2}\):

Now find the domain. Because the expression is under a square root, it must be non-negative:

So the domain is

Why does this happen? The inside function \(g(x)=x+5\) can accept any real number. But the outside function \(f(x)=\sqrt{x-2}\) only accepts inputs greater than or equal to \(2\). Therefore, we need

which gives

This is the domain restriction created by the composition chain.

Worked Example 3: Composition with a rational function

Let

Find \((f \circ g)(x)\) and its domain.

Compose the functions:

Substitute \(2x+3\) into \(f(x)=\frac{1}{x-1}\):

Now find the domain. The denominator cannot be zero:

So the domain is all real numbers except \(-1\).

Notice the reasoning through the composition chain:

• \(g(x)=2x+3\) is defined for all real numbers.
• But \(f\) cannot accept an input of \(1\), because that would make the denominator zero.
• So we must require \(g(x) \ne 1\).

Set that condition:

This gives the same domain restriction.

Worked Example 4: Composition used in a real situation

A tank contains water. Suppose the depth of the water after \(t\) hours is

where depth is measured in centimeters.

Suppose the pressure at depth \(d\) is modeled by

where pressure is measured in suitable units.

To find pressure directly from time, compose the functions:

Now substitute into \(f(d)=2d+10\):

So the composite model is

This means pressure can be found directly from time.

Now think about the realistic domain. Water depth cannot be negative, so:

Also, time cannot be negative, so:

Therefore the meaningful domain is

This example shows why composition is useful in modeling systems that happen in stages.

Composition and inverse functions

Composition is closely related to inverse functions. If \(f\) and \(f^{-1}\) are inverses, then composing them returns the original input:

and

But this only works on the correct domains. Sometimes a function must have its domain restricted to become invertible. For example, \(f(x)=x^2\) is not one-to-one on all real numbers, but if we restrict the domain to \(x \ge 0\), then it has the inverse \(f^{-1}(x)=\sqrt{x}\).

This is another reason domain restrictions matter so much in function composition.

Common mistakes to avoid

• Doing the functions in the wrong order: \((f \circ g)(x)\) means do \(g\) first, then \(f\).
• Forgetting parentheses: When substituting, use brackets carefully. For example, \(f(x)=x^2\) and input \(3x-1\) gives \((3x-1)^2\), not \(3x^2-1\).
• Ignoring domain restrictions: Always check whether the inside output is allowed in the outside function.
• Assuming composition is commutative: In general, \(f \circ g \ne g \circ f\).

A quick strategy for domain checking

1. Write the composite function.
2. Check restrictions from the final simplified expression.
3. Also check restrictions coming from the inside function feeding into the outside function.
4. Keep only values that satisfy all restrictions.

Practice-style check

If \(f(x)=\sqrt{x+1}\) and \(g(x)=\frac{1}{x-2}\), then

To find the domain, we need:

• \(x \ne 2\), because \(g(x)\) is undefined there, and
• \(\frac{1}{x-2}+1 \ge 0\), because the square root needs a non-negative value.

This kind of question shows why composition often requires careful algebra and careful domain analysis together.

Summary

Function composition combines two functions into one by using the output of one function as the input of another. The notation \((f \circ g)(x)=f(g(x))\) means apply \(g\) first, then \(f\).

When composing functions, order matters, substitution must be done carefully, and domain restrictions must be checked through the full chain. In many real-world models, composition helps describe processes that happen in steps, and it is also a key idea for understanding inverse functions.

One-to-One Functions and Invertibility

In function theory, an important question is whether a function can be reversed. If a function takes an input and gives exactly one output, we may ask: can we start with the output and recover exactly one input?

This is only possible when the function is one-to-one. Understanding one-to-one functions helps us know when a function has a true inverse, and when we must restrict the domain to make an inverse possible.

In this lesson, you will learn how to identify one-to-one functions, use the horizontal line test, prove one-to-one behavior algebraically, and understand how domain restrictions make inverse functions possible.

1. What does one-to-one mean?

A function is one-to-one if different inputs always produce different outputs.

In symbols, a function \(f\) is one-to-one if whenever

it must follow that

This means no two different inputs can share the same output.

For example, the function \(f(x)=2x+3\) is one-to-one because every input gives a unique output, and no output is repeated.

But the function \(f(x)=x^2\) is not one-to-one if its domain is all real numbers, because

Two different inputs give the same output, so the function cannot be reversed uniquely over all real numbers.

2. Why one-to-one matters for inverses

An inverse function undoes the original function. If \(f\) sends \(x\) to \(y\), then the inverse sends \(y\) back to \(x\).

For an inverse to be a true function, each output of the original function must come from exactly one input. That is why the original function must be one-to-one.

If a function is not one-to-one, then one output could match more than one input. In that case, the inverse would have to assign one output to multiple inputs, which is not allowed for a function.

So the key fact is:

• If a function is one-to-one, it can have an inverse function.
• If a function is not one-to-one, it does not have an inverse function unless we restrict its domain.

3. The horizontal line test

The horizontal line test is a graph-based way to check whether a function is one-to-one.

A function is one-to-one if and only if every horizontal line intersects its graph at most once.

Why does this work? A horizontal line has the form \(y=c\), where \(c\) is a constant. If the graph is hit more than once by the same horizontal line, then the same output \(c\) comes from more than one input. That means the function is not one-to-one.

Here is how to use it:

1. Look at the graph of the function.
2. Imagine drawing horizontal lines across the graph.
3. If any horizontal line crosses the graph more than once, the function is not one-to-one.
4. If every horizontal line crosses at most once, the function is one-to-one.

Examples from common graphs:

• A line with nonzero slope, like \(y=3x-1\), passes the horizontal line test.
• A parabola like \(y=x^2\) fails the horizontal line test.
• An increasing cubic like \(y=x^3\) passes the horizontal line test.

4. Algebraic test for one-to-one

Sometimes you may not have a graph, or you may need a formal proof. Then you can test one-to-one behavior algebraically.

The method is based on the definition:

1. Start by assuming \(f(a)=f(b)\).
2. Simplify the equation.
3. If you can show that this leads to \(a=b\), then the function is one-to-one.

This is a very important proof method in higher-level function work.

5. Restricting the domain for invertibility

Some functions are not one-to-one on their full domain, but they become one-to-one if we limit the allowed inputs.

This is called restricting the domain.

For example, \(f(x)=x^2\) is not one-to-one for all real numbers. But if we restrict the domain to \(x\ge 0\), then each output has exactly one input in that domain.

On the restricted domain \(x\ge 0\), the function \(f(x)=x^2\) does have an inverse:

We could also restrict the domain to \(x\le 0\). Then the inverse would be

So domain restriction is often the key to making a non-invertible function invertible.

6. One-to-one behavior and function shape

Many one-to-one functions are always increasing or always decreasing over their domain.

If a function always increases as \(x\) increases, then it will not repeat outputs. The same is true if it always decreases.

For example:

• \(f(x)=2x+1\) is always increasing, so it is one-to-one.
• \(f(x)=-x^3\) is always decreasing, so it is one-to-one.
• \(f(x)=x^2\) decreases and then increases, so it repeats outputs and is not one-to-one on all real numbers.

This idea helps you quickly classify function behavior from graphs and equations.

7. Worked Example 1: Using the horizontal line test

Determine whether \(f(x)=x^3\) is one-to-one.

Step 1: Think about the graph of \(y=x^3\).

It is an increasing curve that moves from bottom left to top right.

Step 2: Apply the horizontal line test.

Any horizontal line will meet the graph exactly once.

Conclusion: \(f(x)=x^3\) is one-to-one.

Because it is one-to-one, it has an inverse function.

8. Worked Example 2: Algebraic proof

Show algebraically that \(f(x)=5x-7\) is one-to-one.

Step 1: Assume

Then

Step 2: Solve the equation.

Conclusion: Since \(f(a)=f(b)\) leads to \(a=b\), the function is one-to-one.

9. Worked Example 3: A function that is not one-to-one

Determine whether \(f(x)=x^2-4\) is one-to-one on all real numbers.

Method 1: Using values

Check two different inputs:

Since \(f(3)=f(-3)\) but \(3\ne -3\), the function is not one-to-one.

Method 2: Horizontal line test

The graph is a parabola opening upward. Many horizontal lines intersect it twice.

Conclusion: \(f(x)=x^2-4\) is not one-to-one on all real numbers, so it does not have an inverse function on that full domain.

10. Worked Example 4: Restricting the domain and finding the inverse

Let \(f(x)=x^2-4\), with restricted domain \(x\ge 0\). Show that it is invertible and find its inverse.

Step 1: Explain why the restriction helps.

When \(x\ge 0\), the function \(x^2-4\) is increasing. Each output comes from only one input, so the function is one-to-one on this restricted domain.

Step 2: Write the function as

Step 3: Swap \(x\) and \(y\) to begin finding the inverse.

Step 4: Solve for \(y\).

Step 5: Choose the correct branch.

Because the original domain is \(x\ge 0\), the outputs of the inverse must satisfy \(y\ge 0\). So we choose the positive square root.

Conclusion: After restricting the domain to \(x\ge 0\), the function becomes invertible, and its inverse is

11. How to check if an inverse is correct

You can verify an inverse by composition.

If \(f\) and \(f^{-1}\) are true inverses, then

for values in the appropriate domains.

Using the previous example, let

Check one composition:

So the inverse works correctly.

12. Common mistakes to avoid

• Confusing one-to-one with just being a function: Every one-to-one relation is a function, but not every function is one-to-one.
• Using the vertical line test instead of the horizontal line test: The vertical line test checks whether a graph is a function. The horizontal line test checks whether it is one-to-one.
• Forgetting domain restrictions: A function like \(x^2\) is not invertible on all real numbers, but it can become invertible on a restricted domain.
• Keeping both \(\pm\) signs in an inverse: A true inverse must itself be a function, so you must choose the branch that matches the restricted domain.

13. Quick checklist for one-to-one and invertibility

When you are given a function, ask yourself:

1. Is it a function?
2. Does it pass the horizontal line test?
3. Can I prove algebraically that \(f(a)=f(b)\Rightarrow a=b\)?
4. If it is not one-to-one, can I restrict the domain?
5. After restricting the domain, can I find and check the inverse?

14. Brief summary

A function is one-to-one when different inputs always produce different outputs. This property is necessary for a function to have a true inverse.

You can test one-to-one behavior using the horizontal line test or by an algebraic proof starting with \(f(a)=f(b)\). If a function is not one-to-one on its full domain, you can often restrict the domain to make it invertible.

Understanding these ideas is essential when analyzing relations, building inverse models, and composing functions in more advanced function problems.

Constructing Inverse Functions is about reversing what a function does. If a function takes an input and turns it into an output, then its inverse takes that output and gives back the original input.

For example, if a function doubles a number and then adds 3, its inverse must subtract 3 and then divide by 2. Inverse functions “undo” each other.

In symbols, if a function is called \(f(x)\), its inverse is written \(f^{-1}(x)\). The \(-1\) does not mean reciprocal. It means “inverse function.”

This lesson will show you how to:

• decide when a function has an inverse,
• restrict the domain of a function if needed,
• find the equation of an inverse, and
• check your answer by composition.

1. What does it mean for a function to have an inverse?

A function has an inverse if every output comes from exactly one input. This means the function must be one-to-one.

If two different inputs give the same output, then the inverse would not know which input to return, so the inverse would not be a function.

One way to test whether a graph is one-to-one is the horizontal line test.

• If every horizontal line crosses the graph at most once, the function is one-to-one.
• If some horizontal line crosses more than once, the function is not one-to-one.

For example:

• \(f(x)=2x+5\) is one-to-one, so it has an inverse.
• \(f(x)=x^2\) is not one-to-one over all real numbers, because \(f(2)=4\) and \(f(-2)=4\).

2. The basic steps for constructing an inverse

To find the inverse of a one-to-one function, use these steps:

1. Write the function as \(y=f(x)\).
2. Swap \(x\) and \(y\).
3. Solve for \(y\).
4. Rename \(y\) as \(f^{-1}(x)\).

This works because inverse functions switch inputs and outputs.

Also remember:

• The domain of \(f\) becomes the range of \(f^{-1}\).
• The range of \(f\) becomes the domain of \(f^{-1}\).

3. Worked Example 1: A simple linear function

Find the inverse of \(f(x)=3x-7\).

Step 1: Write the function with \(y\).

$$y=3x-7$$

Step 2: Swap \(x\) and \(y\).

$$x=3y-7$$

Step 3: Solve for \(y\).

Add 7 to both sides:

$$x+7=3y$$

Divide by 3:

$$y=\frac{x+7}{3}$$

Step 4: Write the inverse.

$$f^{-1}(x)=\frac{x+7}{3}$$

Check by composition:

$$f\big(f^{-1}(x)\big)=3\left(\frac{x+7}{3}\right)-7=x+7-7=x$$

$$f^{-1}(f(x))=\frac{(3x-7)+7}{3}=\frac{3x}{3}=x$$

Since both compositions equal \(x\), the inverse is correct.

4. Restricting the domain

Some functions do not have inverses unless we limit their domain. This is called restricting the domain.

The goal is to choose a part of the graph where the function is one-to-one.

A common example is \(f(x)=x^2\). Over all real numbers, it is not one-to-one. But if we restrict the domain to \(x\ge 0\), then each output has exactly one input.

This restriction matches the principal square root, which is the version usually used for inverses.

5. Worked Example 2: Inverse of a quadratic with a restricted domain

Find the inverse of \(f(x)=x^2\) with domain \(x\ge 0\).

Step 1: Write the function with \(y\).

$$y=x^2, \quad x\ge 0$$

Step 2: Swap \(x\) and \(y\).

$$x=y^2$$

Step 3: Solve for \(y\).

$$y=\pm\sqrt{x}$$

But the original function had domain \(x\ge 0\), so its outputs come from nonnegative inputs only. That means the inverse must return only nonnegative values. So we choose:

$$y=\sqrt{x}$$

Step 4: Write the inverse.

$$f^{-1}(x)=\sqrt{x}$$

Important note: If the domain had been restricted to \(x\le 0\), then the inverse would be:

$$f^{-1}(x)=-\sqrt{x}$$

Check by composition:

For \(x\ge 0\),

$$f(f^{-1}(x))=(\sqrt{x})^2=x$$

And

$$f^{-1}(f(x))=\sqrt{x^2}=x$$

This last step works because the original domain was restricted to \(x\ge 0\).

6. Inverses and graphs

The graph of a function and the graph of its inverse are reflections of each other across the line \(y=x\).

This makes sense because inverse functions swap inputs and outputs. A point \((a,b)\) on the graph of \(f\) becomes the point \((b,a)\) on the graph of \(f^{-1}\).

For example, if \((2,11)\) is on \(f(x)=3x+5\), then \((11,2)\) is on \(f^{-1}(x)=\frac{x-5}{3}\).

7. Worked Example 3: A rational function

Find the inverse of \(f(x)=\frac{x-4}{2}\).

Step 1: Write with \(y\).

$$y=\frac{x-4}{2}$$

Step 2: Swap \(x\) and \(y\).

$$x=\frac{y-4}{2}$$

Step 3: Solve for \(y\).

Multiply both sides by 2:

$$2x=y-4$$

Add 4:

$$y=2x+4$$

Step 4: Write the inverse.

$$f^{-1}(x)=2x+4$$

Check:

$$f(f^{-1}(x))=\frac{(2x+4)-4}{2}=\frac{2x}{2}=x$$

This confirms the result.

8. Worked Example 4: A cube root and cube function idea

Find the inverse of \(f(x)=x^3+1\).

This function is increasing for all real numbers, so it is one-to-one and has an inverse without restricting the domain.

Step 1: Write with \(y\).

$$y=x^3+1$$

Step 2: Swap \(x\) and \(y\).

$$x=y^3+1$$

Step 3: Solve for \(y\).

Subtract 1:

$$x-1=y^3$$

Take the cube root:

$$y=\sqrt[3]{x-1}$$

Step 4: Write the inverse.

$$f^{-1}(x)=\sqrt[3]{x-1}$$

Check by composition:

$$f(f^{-1}(x))=\left(\sqrt[3]{x-1}\right)^3+1=x-1+1=x$$

9. Verifying inverses through composition

The best way to confirm an inverse is to compose the functions.

If \(f\) and \(g\) are inverses, then:

$$f(g(x))=x \quad \text{and} \quad g(f(x))=x$$

When you test this, pay attention to domain restrictions. Sometimes a composition only simplifies to \(x\) when \(x\) is in the correct domain.

For example, with square roots and squares:

$$\sqrt{x^2}=|x|$$

not always \(x\).

That is why domain restrictions are so important when constructing inverses.

10. Common mistakes to avoid

• Forgetting to check if the function is one-to-one. Not every function has an inverse over its full domain.
• Thinking \(f^{-1}(x)\) means \(\frac{1}{f(x)}\). It does not. It means the inverse function.
• Not restricting the domain when needed. This is especially important for \(x^2\), trig graphs, and other repeating or symmetric functions.
• Making sign errors when solving for \(y\). Work carefully through each algebra step.
• Skipping the composition check. A quick check can catch mistakes.

11. Quick process you can use on any problem

1. Ask: Is the function one-to-one?
2. If not, restrict the domain to a one-to-one part.
3. Replace \(f(x)\) with \(y\).
4. Swap \(x\) and \(y\).
5. Solve for \(y\).
6. Write \(f^{-1}(x)\).
7. Check using composition.

Summary

To construct an inverse function, the original function must be one-to-one, or its domain must be restricted so that it becomes one-to-one. Then you swap \(x\) and \(y\), solve for \(y\), and write the result as \(f^{-1}(x)\).

Inverse functions undo each other, and you can verify them by showing that both compositions equal \(x\). Always pay close attention to domain and range, especially for functions like \(x^2\) that need a restricted domain before an inverse can exist.