Functions, Mappings, and Transformations

Lesson: Relations vs. Functions

In math, we often study how one value is connected to another value. These connections are called relations. A relation can match inputs with outputs in many different ways.

A function is a special kind of relation. In a function, each input must match with exactly one output. This is the most important idea in this lesson.

If you remember one rule, remember this: a relation is any set of ordered pairs, but a function is a relation where no input is paired with more than one output.

1. What is a relation?

A relation is any set of ordered pairs. An ordered pair looks like \\((x, y)\\), where:

• the first number, \\(x\\), is the input
• the second number, \\(y\\), is the output

For example, the set

$$\{(1,2), (2,4), (3,6)\}$$

is a relation because it is a set of ordered pairs.

A relation can be shown in different ways:

• as ordered pairs
• in a table
• on a graph
• with a mapping diagram

2. What is a function?

A function is a relation with a stricter rule. Every input must have one and only one output.

This means:

• an input can have one output
• different inputs can share the same output
• but one input cannot have two different outputs

For example,

$$\{(1,5), (2,5), (3,8)\}$$

is a function. Even though both \\(1\\) and \\(2\\) go to \\(5\\), that is allowed because each input still has only one output.

But this set is not a function:

$$\{(1,5), (1,7), (2,8)\}$$

It is not a function because the input \\(1\\) has two outputs: \\(5\\) and \\(7\\).

3. Domain and range

To understand relations and functions, it helps to know two important words:

• Domain: the set of all inputs
• Range: the set of all outputs

For the relation

$$\{(2,3), (4,7), (6,7)\}$$

the domain is \\(\{2,4,6\}\\) and the range is \\(\{3,7\}\\).

Notice that the range does not repeat values. Even though \\(7\\) appears twice as an output, we list it once in the set.

4. How to tell if a relation is a function

There are several ways to check.

A. From ordered pairs

Look at the inputs. If any input is repeated with a different output, then the relation is not a function.

B. From a table

Check the input column. If the same input appears more than once with different outputs, it is not a function.

C. From a mapping diagram

Each input should have exactly one arrow going out. If one input points to two different outputs, it is not a function.

D. From a graph

Use the vertical line test. If a vertical line touches the graph more than once, then the graph is not a function.

This works because a vertical line represents one input value \\(x\\). If that line hits the graph twice, then that one input has two outputs.

5. Relation vs. function

Here is the key difference:

• Relation: any pairing of inputs and outputs
• Function: a relation where each input has exactly one output

So, every function is a relation, but not every relation is a function.

6. Worked Examples

Example 1: Ordered pairs

Decide whether this relation is a function:

$$\{(2,1), (3,4), (4,9), (5,16)\}$$

Step 1: Look at the inputs: \\(2, 3, 4, 5\\).

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

No input repeats.

Answer: Yes, this relation is a function.

Example 2: Repeated input

Decide whether this relation is a function:

$$\{(1,3), (2,5), (1,7), (4,9)\}$$

Step 1: Look at the inputs: \\(1, 2, 1, 4\\).

Step 2: Notice that the input \\(1\\) appears twice.

Step 3: Check the outputs for that input. The input \\(1\\) is paired with \\(3\\) and \\(7\\).

Since one input has two different outputs, the relation breaks the function rule.

Answer: No, this relation is not a function.

Example 3: Table of values

Use the table to decide if the relation is a function.

$$ \begin{array}{c|c} x & y \\\ \hline -1 & 2 \\ 0 & 5 \\ 1 & 2 \\ 2 & 8 \end{array} $$

Step 1: Look at the input values: \\(-1, 0, 1, 2\\).

Step 2: Each input appears only once.

Step 3: Notice that the output \\(2\\) appears twice. That is okay, because different inputs can have the same output.

Answer: Yes, the relation is a function.

Example 4: Thinking about a graph

Suppose a graph is a circle. Is it a function?

Imagine drawing a vertical line through the middle of the circle. That line would touch the circle at two points.

That means one input value has two output values.

Answer: A circle is not a function.

7. Common mistakes to avoid

• Mistake 1: Thinking repeated outputs mean it is not a function. Repeated outputs are allowed.
• Mistake 2: Looking at outputs instead of inputs. The function rule is about inputs having exactly one output.
• Mistake 3: Forgetting that a repeated input with the same output is still okay in many school examples of relations, because it does not change the output. But usually in a set of ordered pairs, repeated identical pairs are not listed more than once.
• Mistake 4: On graphs, forgetting to use vertical lines. The vertical line test checks whether one \\(x\\)-value has more than one \\(y\\)-value.

8. Quick check questions

Try these on your own:

1. Is \\(\{(3,1), (4,2), (5,3)\}\\) a function?
2. Is \\(\{(2,6), (2,7), (3,8)\}\\) a function?
3. If a mapping diagram shows one input pointing to two outputs, is it a function?
4. If two different inputs point to the same output, is it still a function?

Answers:

1. Yes
2. No
3. No
4. Yes

9. Summary

A relation is any set of ordered pairs that connects inputs and outputs. A function is a special relation where each input has exactly one output.

To decide if something is a function, focus on the inputs. If one input is matched with more than one output, it is not a function. If every input has only one output, then it is a function.

Remember: all functions are relations, but not all relations are functions.

Domain and Range Definitions

When we work with functions, we often ask two important questions:

• What inputs are allowed?
• What outputs can happen?

The answers to these questions are called the domain and the range.

Understanding domain and range helps you read equations, graphs, and real-world situations more clearly. It tells you what values make sense to use and what results are possible.

Definition of Domain

The domain of a function is the set of all possible input values. These are usually the values of \(x\) that you are allowed to use.

For example, in the function \(y = 2x + 5\), you can put in any real number for \(x\). So the domain is all real numbers.

Definition of Range

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

For the same function \(y = 2x + 5\), if \(x\) can be any real number, then \(y\) can also be any real number. So the range is all real numbers.

Think of it like a machine:

• The domain is everything you are allowed to put into the machine.
• The range is everything that can come out of the machine.

How to Find Domain and Range

You can find domain and range in different ways, depending on how the function is shown.

1. From an equation — look for values that are not allowed.
2. From a graph — look at how far the graph goes left and right for domain, and up and down for range.
3. From a real-world situation — think about what inputs and outputs make sense in context.

1. Finding Domain from an Equation

Many functions allow all real numbers as inputs. But sometimes certain values are not allowed.

Here are common restrictions:

• You cannot divide by 0.
• You usually do not take the square root of a negative number in 9th Grade math.

So when finding domain, always ask:

• Will any input make the denominator 0?
• Will any input make the value inside a square root negative?

2. Finding Range from an Equation

To find the range, think about what output values are possible.

Sometimes this is easy:

• In a linear function like \(y = 3x - 1\), the outputs can be any real number.

Sometimes the equation has a lowest or highest value:

• In \(y = x^2\), the smallest output is \(0\), because squaring any real number never gives a negative result.

So the range of \(y = x^2\) is all real numbers greater than or equal to \(0\).

3. Finding Domain and Range from a Graph

On a graph:

• The domain is the set of all \(x\)-values covered by the graph.
• The range is the set of all \(y\)-values covered by the graph.

A good way to remember this is:

• Domain = look left to right.
• Range = look bottom to top.

If the graph continues forever, the domain or range may be all real numbers or continue without end in one direction.

4. Finding Domain and Range in Context

In real life, not every number makes sense.

For example, if \(x\) represents the number of hours worked, then:

• \(x\) cannot be negative.
• You may only use values that make sense for the situation.

If \(y\) represents money earned, then the range also depends on the situation.

This means domain and range are not only about math rules. They are also about reasonable values in a given problem.

Ways to Write Domain and Range

You may see domain and range written in words or using inequalities.

Examples:

• All real numbers
• \(x \geq 0\)
• \(y \geq 2\)
• \(-3 \leq x \leq 5\)

At this level, it is perfectly fine to describe them clearly in words, especially when learning.

Worked Example 1: Linear Function

Find the domain and range of:

$$y = 4x - 7$$

Step 1: Find the domain.

This is a linear equation. There is no denominator and no square root. That means there are no restrictions on \(x\).

So the domain is all real numbers.

Step 2: Find the range.

A line like this continues forever, and as \(x\) changes, \(y\) can also become any real number.

So the range is all real numbers.

Answer:

• Domain: all real numbers
• Range: all real numbers

Worked Example 2: Quadratic Function

Find the domain and range of:

$$y = x^2$$

Step 1: Find the domain.

You can square any real number. There are no values of \(x\) that are not allowed.

So the domain is all real numbers.

Step 2: Find the range.

When you square a number, the result is never negative.

• If \(x = 0\), then \(y = 0\).
• If \(x = 2\), then \(y = 4\).
• If \(x = -2\), then \(y = 4\).

The smallest possible value of \(y\) is \(0\).

So the range is:

$$y \geq 0$$

Answer:

• Domain: all real numbers
• Range: \(y \geq 0\)

Worked Example 3: Fraction Function

Find the domain and range of:

$$y = \frac{1}{x}$$

Step 1: Find the domain.

You cannot divide by 0. So \(x = 0\) is not allowed.

That means the domain is all real numbers except 0.

Step 2: Find the range.

Think about the outputs of \(\frac{1}{x}\).

• If \(x = 1\), then \(y = 1\).
• If \(x = 2\), then \(y = \frac{1}{2}\).
• If \(x = -1\), then \(y = -1\).

The output can be positive or negative, but it can never be \(0\), because \(\frac{1}{x}\) is never 0.

So the range is all real numbers except 0.

Answer:

• Domain: all real numbers except 0
• Range: all real numbers except 0

Worked Example 4: Real-World Situation

A taxi charges a starting fee of \(\$3\) and then \(\$2\) for each mile traveled. Let \(m\) be the number of miles, and let \(C\) be the total cost.

The function is:

$$C = 2m + 3$$

Step 1: Find the domain.

The number of miles cannot be negative. So:

$$m \geq 0$$

In context, the domain is all real numbers greater than or equal to 0.

Step 2: Find the range.

If \(m = 0\), then:

$$C = 2(0) + 3 = 3$$

As miles increase, the cost increases. The lowest cost is \(\$3\).

So the range is:

$$C \geq 3$$

Answer:

• Domain: \(m \geq 0\)
• Range: \(C \geq 3\)

Common Mistakes to Avoid

• Mixing up domain and range — domain is input, range is output.
• Forgetting restrictions — check for division by 0 and square roots of negative numbers.
• Ignoring the context — in real life, negative values or impossible values may not make sense.
• Looking at only one part of a graph — make sure to consider the entire graph.

Quick Check Questions

Try these on your own:

1. What is the domain of \(y = x + 8\)?
2. What is the range of \(y = x^2 + 1\)?
3. Why is \(x = 0\) not in the domain of \(y = \frac{5}{x}\)?
4. If \(t\) is time in hours, why would \(t < 0\) usually not make sense in a real-world problem?

Brief Summary

The domain is the set of all allowed inputs, and the range is the set of all possible outputs. You can find them from equations, graphs, or real-world situations.

Always check whether any inputs are not allowed, and think about what outputs the function can actually produce. If the problem is about a real situation, make sure the values make sense in context.

Vertical Line Test Application

When you look at a graph, one important question is: Does this graph represent a function? The vertical line test is a quick visual way to answer that question.

In 9th Grade Maths, a function is a rule that gives exactly one output for each input. On a graph, this means each -value can match with only one y-value.

The vertical line test helps us check this idea using the graph itself.

What is the vertical line test?

The vertical line test says:

If any vertical line crosses a graph more than once, then the graph is not a function.

If every vertical line crosses the graph at most once, then the graph is a function.

This works because a vertical line has a fixed x-value. If that same x-value hits the graph in two or more places, then one input has more than one output. That breaks the definition of a function.

For example, if a graph includes both points \((2,3)\) and \((2,-1)\), then when the input is \(2\), there are two different outputs: \(3\) and \(-1\). That is not a function.

Why are the lines vertical?

We use vertical lines because they test one x-value at a time. Since functions are about each input having exactly one output, we must check whether any single input is matched with more than one output.

A horizontal line would test whether one y-value has more than one x-value, but that is a different question. Many functions can have the same output for different inputs, so the horizontal line test is not used to decide whether something is a function.

How to apply the vertical line test

1. Look at the graph.

2. Imagine sliding a vertical line from left to right across the graph.

3. Check how many times the line touches the graph at each position.

4. If the line ever touches the graph more than once, the graph is not a function.

5. If this never happens, the graph is a function.

Main idea to remember: one input, one output.

Graphs that usually pass the test

• A straight line like \(y=2x+1\)

• A parabola opening up or down like \(y=x^2\)

• A V-shaped graph like \(y=|x|\)

Even though some of these graphs curve or turn, each x-value still gives only one y-value.

Graphs that usually fail the test

• A circle like \(x^2+y^2=9\)

• A sideways parabola like \(x=y^2\)

• Any graph where a vertical line hits two or more points

These fail because some x-values match with more than one y-value.

Worked Example 1: A line

Decide whether the graph of \(y=3x-2\) is a function.

Step 1: Think about its graph. It is a straight line.

Step 2: Imagine drawing vertical lines through the graph.

Step 3: Each vertical line crosses the graph only once.

Conclusion: The graph is a function.

This makes sense because every input has exactly one output.

Worked Example 2: A circle

Decide whether the graph of

$$x^2+y^2=16$$

is a function.

This graph is a circle centered at the origin with radius \(4\).

Now test the vertical line \(x=0\). It crosses the circle at two points:

$$ (0,4) \text{ and } (0,-4) $$

That means the input \(0\) gives two outputs, \(4\) and \(-4\).

Conclusion: The graph is not a function.

Worked Example 3: A parabola

Decide whether the graph of

$$y=x^2$$

is a function.

This parabola opens upward.

If you draw a vertical line at any x-value, it hits the graph only once. For example:

• At \(x=1\), the graph has the point \((1,1)\)

• At \(x=-2\), the graph has the point \((-2,4)\)

No single x-value gives two different y-values.

Conclusion: The graph is a function.

Worked Example 4: A sideways parabola

Decide whether the graph of

$$x=y^2$$

is a function.

This graph opens to the right.

Try the vertical line \(x=4\). Then

$$4=y^2$$

So

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

The vertical line \(x=4\) hits the graph at \((4,2)\) and \((4,-2)\).

One input gives two outputs.

Conclusion: The graph is not a function.

Common mistakes

• Mistake 1: Thinking that a graph must be a straight line to be a function. This is false. Many curved graphs are functions.

• Mistake 2: Looking at whether the graph goes up or down. A graph can rise and fall and still be a function.

• Mistake 3: Using a horizontal line test instead of a vertical line test. To check if a graph is a function, always use vertical lines.

• Mistake 4: Forgetting that just one vertical line crossing twice is enough to show it is not a function.

A quick way to think about it

Ask yourself: Can one x-value land on more than one point of the graph?

If the answer is yes, then the graph is not a function.

If the answer is no, then the graph is a function.

Connection to mappings and input-output rules

In a mapping diagram, a function cannot send one input to two different outputs. The vertical line test shows the same idea on a graph.

So the graph, the table, the equation, and the mapping diagram all follow the same rule:

$$\text{Each input has exactly one output.}$$

Summary

The vertical line test is a visual method for deciding whether a graph represents a function. If any vertical line crosses the graph more than once, the graph is not a function. If every vertical line crosses at most once, the graph is a function. Always connect the test back to the meaning of a function: one input, one output.

Function Notation Evaluation is a way of writing and understanding rules that connect an input to an output. In 9th Grade Maths, this helps you read functions clearly and substitute values correctly.

A function takes an input, does something to it, and gives exactly one output. Function notation is usually written like \(f(x)\), which is read as “f of x.”

It is very important to know that \(f(x)\) does not mean \(f \times x\). Instead, it names the output of a function when the input is \(x\).

For example, if a function is defined by

$$f(x)=2x+3$$

this means: take the input, multiply it by 2, then add 3.

If the input is 4, then we write \(f(4)\). This means “find the output when \(x=4\).”

$$f(4)=2(4)+3=8+3=11$$

So the output is 11.

Function notation helps us keep track of which input we are using. The number inside the parentheses is the input. The result after substituting and simplifying is the output.

Main Idea: To evaluate a function, replace every \(x\) in the rule with the given value, then simplify.

Here is the basic process:

1. Write the function rule.
2. Replace \(x\) with the given input.
3. Use parentheses carefully.
4. Simplify step by step.

Let’s look at this with a simple example.

Worked Example 1: Evaluate \(f(5)\) if

$$f(x)=x+7$$

Substitute 5 for \(x\):

$$f(5)=5+7=12$$

So, \(f(5)=12\).

This means the function takes the input 5 and gives the output 12.

Worked Example 2: Evaluate \(g(-2)\) if

$$g(x)=3x-4$$

Replace \(x\) with \(-2\):

$$g(-2)=3(-2)-4$$ $$=-6-4=-10$$

So, \(g(-2)=-10\).

Notice how the negative number was placed in parentheses. This helps avoid mistakes.

When substituting negative numbers, always use parentheses. For example, writing \(3-2\) instead of \(3(-2)\) would completely change the meaning.

Worked Example 3: Evaluate \(h(3)\) if

$$h(x)=x^2-5x+1$$

Substitute 3 for \(x\):

$$h(3)=3^2-5(3)+1$$

Simplify in order:

$$=9-15+1$$ $$=-6+1=-5$$

So, \(h(3)=-5\).

This example shows that evaluating a function may involve more than one operation. Substitute first, then simplify carefully.

Sometimes the input is not a number. It could be another expression.

Worked Example 4: Evaluate \(f(a+1)\) if

$$f(x)=2x-3$$

Replace every \(x\) with \(a+1\):

$$f(a+1)=2(a+1)-3$$

Now simplify:

$$=2a+2-3$$ $$=2a-1$$

So, \(f(a+1)=2a-1\).

This is still function evaluation. The input does not have to be a single number. You follow the same rule: substitute the entire input wherever \(x\) appears.

Important Things to Remember

• \(f(x)\) means the output of function \(f\) for input \(x\).
• \(f(2)\) means substitute 2 for every \(x\).
• \(f(-3)\) means substitute \(-3\), using parentheses.
• The letter name can change. You might see \(f(x)\), \(g(x)\), or \(h(x)\). They are just different function names.

For example, if

$$g(x)=x^2+1$$

then

$$g(4)=4^2+1=16+1=17$$

The letter \(g\) is just the name of the function. The process is still the same.

Common Mistakes

• Thinking \(f(x)\) means multiplication: It does not mean \(f \times x\). It names a function output.
• Forgetting parentheses with negatives: For \(f(-2)\), substitute \((-2)\), not just \(-2\) without care.
• Not replacing every \(x\): Every \(x\) in the rule must be changed to the given input.
• Ignoring order of operations: After substituting, simplify carefully.

Here is a quick comparison:

If

$$f(x)=x^2$$

then

$$f(-3)=(-3)^2=9$$

But if you forget the parentheses and write

$$-3^2=-9$$

you get the wrong answer. The parentheses show that the input itself is negative 3.

How Function Notation Connects to Inputs and Outputs

You can think of a function like a machine:

• Input goes in.
• The rule acts on the input.
• Output comes out.

For the function

$$f(x)=2x+3$$

an input of 1 gives:

$$f(1)=2(1)+3=5$$

an input of 6 gives:

$$f(6)=2(6)+3=15$$

Different inputs can give different outputs, but each input has exactly one output in a function.

Try These Mentally

• If \(p(x)=x-8\), then \(p(10)=2\).
• If \(m(x)=4x\), then \(m(7)=28\).
• If \(q(x)=x^2+2\), then \(q(0)=2\).

In each case, you are simply replacing \(x\) with the given input.

Summary

Function notation tells you the output of a rule for a given input. To evaluate a function, substitute the input value into the function rule and simplify. Always use parentheses when substituting negative numbers or expressions. If you follow the substitution carefully, function notation becomes a clear way to track inputs and outputs.

Linear vs. Nonlinear Growth Rates

In math, many situations show how one quantity changes as another quantity changes. For example, the total cost of tickets changes as the number of tickets changes, or the height of a plant changes as time passes.

A big idea in 9th Grade math is understanding growth rate. A growth rate tells us how much the output changes when the input changes. This helps us decide whether a function is linear or nonlinear.

In this lesson, you will learn how to recognize linear and nonlinear growth rates using tables, graphs, and equations.

1. What is a linear growth rate?

A function has a linear growth rate when it changes by the same amount each time. In other words, the rate of change is constant.

If the input increases by 1, and the output always increases by the same number, then the function is linear.

Linear functions can often be written in the form

$$y = mx + b$$

where:

• \(m\) is the constant rate of change, also called the slope
• \(b\) is the starting value, also called the y-intercept

2. What is a nonlinear growth rate?

A function is nonlinear when the rate of change is not constant. This means the output does not increase or decrease by the same amount each time the input changes equally.

For example, if the input goes up by 1 but the output changes by different amounts, then the function is nonlinear.

Nonlinear functions can have many different shapes and patterns. Some curves rise slowly at first and then faster. Others may rise and fall. The key idea is that the change is not constant.

3. How to tell from a table

When looking at a table, first check whether the input values change by equal amounts. If they do, then compare the changes in the output values.

If the output changes by the same amount each time, the function is linear. If not, it is nonlinear.

Example of a linear table:

$$\begin{array}{c|c} x & y \\\hline 1 & 4 \\ 2 & 7 \\ 3 & 10 \\ 4 & 13 \end{array}$$

As \(x\) increases by 1 each time, \(y\) increases by 3 each time:

$$4 \to 7 \to 10 \to 13$$

Since the output change is always \(+3\), this function is linear.

Example of a nonlinear table:

$$\begin{array}{c|c} x & y \\\hline 1 & 2 \\ 2 & 5 \\ 3 & 10 \\ 4 & 17 \end{array}$$

Again, \(x\) increases by 1 each time, but now the changes in \(y\) are:

$$2 \to 5 \text{ is } +3, \quad 5 \to 10 \text{ is } +5, \quad 10 \to 17 \text{ is } +7$$

Because the output does not change by the same amount each time, this function is nonlinear.

4. How to tell from a graph

A linear function graphs as a straight line. This happens because the rate of change stays constant.

A nonlinear function does not graph as a straight line. It may curve, bend, or change direction.

So, when looking at a graph, ask:

• Is it a straight line? Then it is linear.
• Is it curved or not straight? Then it is nonlinear.

5. How to tell from an equation

A linear equation in 9th Grade usually looks like

$$y = mx + b$$

In this form, the variable has an exponent of 1, and the terms do not create curves.

Examples of linear equations:

• \(y = 2x + 5\)
• \(y = -3x + 1\)
• \(y = 7 - x\)

Examples of nonlinear equations:

• \(y = x^2 + 1\)
• \(y = x^2 - 4x\)
• \(y = 2^x\)

These are nonlinear because they do not have a constant rate of change.

6. Worked Example 1: Using a table

Decide whether the function is linear or nonlinear.

$$\begin{array}{c|c} x & y \\\hline 0 & 1 \\ 1 & 4 \\ 2 & 7 \\ 3 & 10 \end{array}$$

Step 1: Check how \(x\) changes.

The input increases by 1 each time.

Step 2: Check how \(y\) changes.

$$1 \to 4 \text{ is } +3, \quad 4 \to 7 \text{ is } +3, \quad 7 \to 10 \text{ is } +3$$

Conclusion: The output changes by a constant amount, so the function is linear.

7. Worked Example 2: Using a table with changing growth

Decide whether the function is linear or nonlinear.

$$\begin{array}{c|c} x & y \\\hline 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \end{array}$$

Step 1: The input increases by 1 each time.

Step 2: Find the changes in the output.

$$0 \to 1 \text{ is } +1, \quad 1 \to 4 \text{ is } +3, \quad 4 \to 9 \text{ is } +5$$

Conclusion: The changes are not constant, so the function is nonlinear.

8. Worked Example 3: Using an equation

Decide whether \(y = -2x + 6\) is linear or nonlinear.

Step 1: Compare it to the linear form \(y = mx + b\).

Here, \(m = -2\) and \(b = 6\).

Step 2: Since it matches the form \(y = mx + b\), it has a constant rate of change.

Conclusion: This function is linear.

Its growth rate is actually negative, which means the function decreases by 2 each time \(x\) increases by 1.

9. Worked Example 4: Using a graph description

Suppose a graph rises in a curve instead of a straight line. Is it linear or nonlinear?

Step 1: Remember that linear graphs are straight lines.

Step 2: A curved graph does not have a constant rate of change.

Conclusion: The function is nonlinear.

10. Comparing linear and nonlinear growth

• Linear growth: adds or subtracts the same amount over equal input changes
• Nonlinear growth: changes by different amounts over equal input changes
• Linear graph: straight line
• Nonlinear graph: not a straight line
• Linear equation: usually can be written as \(y = mx + b\)
• Nonlinear equation: does not fit that form

11. A useful check: rate of change

If you are unsure, always return to this question:

When the input changes equally, does the output change equally?

If yes, the function is linear.

If no, the function is nonlinear.

12. Common mistakes to avoid

• Looking only at the first two values: You must check all the changes in the table.
• Confusing increasing with linear: Just because a function goes up does not mean it is linear. It must go up by the same amount each time.
• Ignoring negative rate of change: A function can still be linear if it decreases at a constant rate.
• Assuming every equation is linear: Only equations with constant rate of change are linear.

13. Quick practice questions

1. Is this table linear or nonlinear?

   $$\begin{array}{c|c} x & y \\\hline 1 & 3 \\ 2 & 6 \\ 3 & 9 \\ 4 & 12 \end{array}$$

2. Is \(y = x^2 + 2\) linear or nonlinear?

3. If a graph is a straight line going downward, is it linear or nonlinear?

4. If the outputs change by \(+2, +2, +2\), is the function linear or nonlinear?

Answers:

1. Linear
2. Nonlinear
3. Linear
4. Linear

Summary

A function is linear if it has a constant rate of change. In a table, this means the outputs change by the same amount when the inputs change equally. On a graph, a linear function is a straight line.

A function is nonlinear if its rate of change is not constant. In a table, the output changes are different. On a graph, it will not be a straight line.

Whenever you need to decide between linear and nonlinear, look for one key idea: constant change or not?

Piecewise-Defined Functions are functions that use different rules for different input values. Instead of one formula working for every value of \(x\), the function is split into parts. Each part tells you which formula to use for a certain interval or condition.

This idea is useful in real life. For example, a parking garage might charge one rate for the first hour and a different rate after that. A delivery service might charge one amount for short distances and another for longer distances. In math, piecewise functions let us describe these changing rules clearly.

A piecewise-defined function is usually written like this:

$$ f(x)= \begin{cases} \text{rule 1} & \text{if condition 1} \\ \text{rule 2} & \text{if condition 2} \\ \text{rule 3} & \text{if condition 3} \end{cases} $$

Each line gives:

• a formula or value to use, and
• the values of \(x\) for which that formula applies.

The most important skill is to match the input value to the correct condition. Once you choose the right piece, evaluate it like any other expression.

Example of a piecewise function:

$$ f(x)= \begin{cases} 2x+1 & \text{if } x<0 \\ x^2 & \text{if } x\ge 0 \end{cases} $$

This means:

• If \(x\) is negative, use \(2x+1\).
• If \(x\) is zero or positive, use \(x^2\).

How to evaluate a piecewise function

1. Look at the input value.
2. Decide which condition the input satisfies.
3. Use the formula from that piece only.
4. Simplify to get the output.

Worked Example 1: Evaluating a piecewise function

Let

$$ g(x)= \begin{cases} 3x-2 & \text{if } x<1 \\ 5 & \text{if } x=1 \\ x+4 & \text{if } x>1 \end{cases} $$

Find \(g(-2)\), \(g(1)\), and \(g(3)\).

Step 1: Find \(g(-2)\)

Since \(-2<1\), use \(3x-2\).

$$ g(-2)=3(-2)-2=-6-2=-8 $$

So, \(g(-2)=-8\).

Step 2: Find \(g(1)\)

Since \(x=1\), use the middle piece.

$$ g(1)=5 $$

So, \(g(1)=5\).

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

Since \(3>1\), use \(x+4\).

$$ g(3)=3+4=7 $$

So, \(g(3)=7\).

Important note: You only use one piece for each input. Do not mix formulas together.

Understanding the conditions

The conditions tell you which values belong to each piece. Pay close attention to symbols such as:

• \(<\) means less than
• \(>\) means greater than
• \(\le\) means less than or equal to
• \(\ge\) means greater than or equal to

These symbols matter because they decide whether an endpoint is included.

For example, in

$$ h(x)= \begin{cases} x+2 & \text{if } x\le 3 \\ 2x-1 & \text{if } x>3 \end{cases} $$

the value \(x=3\) belongs to the first piece because of the \(\le 3\).

Worked Example 2: Evaluating at boundary values

Let

$$ h(x)= \begin{cases} x^2 & \text{if } x\le 2 \\ 2x+1 & \text{if } x>2 \end{cases} $$

Find \(h(2)\) and \(h(4)\).

Find \(h(2)\)

Since \(2\le 2\), use \(x^2\).

$$ h(2)=2^2=4 $$

Find \(h(4)\)

Since \(4>2\), use \(2x+1\).

$$ h(4)=2(4)+1=8+1=9 $$

So, \(h(2)=4\) and \(h(4)=9\).

Graphing piecewise-defined functions

To graph a piecewise function, graph each rule only on the interval where it is allowed. This is different from graphing a formula over all values of \(x\).

When graphing, endpoints are very important:

• Use a closed dot if the endpoint is included, such as with \(\le\) or \(\ge\).
• Use an open dot if the endpoint is not included, such as with \(<\) or \(>\).

Steps for graphing a piecewise function

1. Read one piece at a time.
2. Graph the rule only for the given interval.
3. Check each endpoint.
4. Draw an open or closed dot at the endpoint if needed.
5. Repeat for the other pieces.

Worked Example 3: Graphing a simple piecewise function

Graph

$$ f(x)= \begin{cases} x+1 & \text{if } x<2 \\ 4 & \text{if } x\ge 2 \end{cases} $$

First piece: \(y=x+1\) for \(x<2\)

This is a line with slope 1. But we only draw it for values less than 2.

At \(x=2\), the line would give:

$$ y=2+1=3 $$

Because the condition is \(x<2\), \(x=2\) is not included. So place an open dot at \((2,3)\).

Second piece: \(y=4\) for \(x\ge 2\)

This is a horizontal line at height 4, starting at \(x=2\) and going right.

Because the condition is \(x\ge 2\), \(x=2\) is included. So place a closed dot at \((2,4)\).

The final graph has:

• part of the line \(y=x+1\) to the left of \(x=2\), with an open dot at \((2,3)\)
• a horizontal line \(y=4\) for \(x\ge 2\), with a closed dot at \((2,4)\)

Why open and closed dots matter

Open and closed dots show whether the function actually includes a point. A graph may seem connected or almost connected, but the dot tells the truth about the exact function value at that input.

For a function, each input can have only one output. So if two pieces meet at the same \(x\)-value, only one of them can include that endpoint.

Worked Example 4: Table and graph thinking

Let

$$ p(x)= \begin{cases} -2 & \text{if } x<0 \\ x+1 & \text{if } 0\le x<3 \\ 5-x & \text{if } x\ge 3 \end{cases} $$

Find several points and describe the graph.

For \(x<0\): the output is always \(-2\).

• \(p(-3)=-2\)
• \(p(-1)=-2\)

For \(0\le x<3\): use \(x+1\).

• \(p(0)=0+1=1\)
• \(p(1)=1+1=2\)
• \(p(2)=2+1=3\)

For \(x\ge 3\): use \(5-x\).

• \(p(3)=5-3=2\)
• \(p(4)=5-4=1\)

Describe the graph:

• For negative \(x\)-values, the graph is a horizontal line at \(y=-2\).
• At \(x=0\), the middle piece begins with a closed dot at \((0,1)\).
• The graph follows the line \(y=x+1\) until \(x=3\), but does not include \(x=3\), so there is an open dot at \((3,4)\).
• At \(x=3\), the last piece starts with a closed dot at \((3,2)\).
• Then the graph follows the line \(y=5-x\) for values greater than 3.

Reading piecewise graphs

Sometimes you may be given the graph instead of the formula. To read the function value from a graph:

1. Find the input on the \(x\)-axis.
2. Move up or down to the point on the graph.
3. Read the corresponding \(y\)-value.
4. If there is an open dot at that point, that point is not included.
5. If there is also a closed dot at the same \(x\)-value, use the closed dot's \(y\)-value.

Common mistakes to avoid

• Using the wrong piece: Always check the condition first.
• Forgetting the equals sign: \(x=2\) belongs to \(x\le 2\), not \(x<2\).
• Using more than one rule: Each input uses only one formula.
• Graphing the whole line: Only graph each rule on its assigned interval.
• Mixing up open and closed dots: Open means not included; closed means included.

Quick check for understanding

Suppose

$$ q(x)= \begin{cases} 2x & \text{if } x<3 \\ 7 & \text{if } x\ge 3 \end{cases} $$

• \(q(1)=2(1)=2\)
• \(q(3)=7\) because \(3\ge 3\)
• \(q(5)=7\)

At \(x=3\), the first piece would give \(2(3)=6\), but that point is not included because the condition is \(x<3\). So the graph has an open dot at \((3,6)\) and a closed dot at \((3,7)\).

Summary

A piecewise-defined function uses different formulas for different parts of the domain. To evaluate it, first choose the correct piece by checking the input value. To graph it, draw each rule only on its interval and use open or closed dots to show whether endpoints are included.

Once you get used to reading the conditions carefully, piecewise functions become much easier. The key is always the same: match the input to the correct rule.

Step Functions are functions whose graphs look like stairs. Instead of changing smoothly, the output stays the same for a while and then suddenly jumps to a new value.

These functions are useful when a quantity changes in fixed steps rather than continuously. For example, the number of taxis needed for a group, the number of boxes needed to pack items, or the cost of parking charged by the hour can all be modeled with step functions.

In this lesson, you will learn what a step function is, how to read and sketch one, and how the greatest integer function, floor function, and ceiling function are used in real situations.

1. What is a step function?

A step function is a function made of horizontal pieces. On each interval of input values, the function has one constant output. Then at certain points, the output jumps to a different value.

This means a step function is not continuous. There are breaks or jumps in the graph.

For example, imagine a parking lot that charges:

• $2 for up to 1 hour,
• $4 for more than 1 hour and up to 2 hours,
• $6 for more than 2 hours and up to 3 hours.

The cost does not slowly increase every minute. It stays the same for a whole hour, then jumps up.

2. The floor function

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

In simpler words, \(\lfloor x \rfloor\) means round down to the nearest integer.

Examples:

• \(\lfloor 3.8 \rfloor = 3\)
• \(\lfloor 5 \rfloor = 5\)
• \(\lfloor 0.2 \rfloor = 0\)
• \(\lfloor -1.4 \rfloor = -2\)

That last example is important. For negative numbers, rounding down means going to the smaller integer. Since \(-2 < -1.4\), we have \(\lfloor -1.4 \rfloor = -2\).

The graph of \(y = \lfloor x \rfloor\) has horizontal steps:

• for \(0 \le x < 1\), \(y = 0\)
• for \(1 \le x < 2\), \(y = 1\)
• for \(2 \le x < 3\), \(y = 2\)

And similarly on the negative side:

• for \(-1 \le x < 0\), \(y = -1\)
• for \(-2 \le x < -1\), \(y = -2\)

Each step includes the left endpoint and does not include the right endpoint.

3. The ceiling function

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

In simpler words, \(\lceil x \rceil\) means round up to the nearest integer.

Examples:

• \(\lceil 3.2 \rceil = 4\)
• \(\lceil 7 \rceil = 7\)
• \(\lceil 0.01 \rceil = 1\)
• \(\lceil -1.4 \rceil = -1\)

The graph of \(y = \lceil x \rceil\) is also a step function, but the steps are arranged a little differently:

• for \(0 < x \le 1\), \(y = 1\)
• for \(1 < x \le 2\), \(y = 2\)
• for \(-1 < x \le 0\), \(y = 0\)

Each step does not include the left endpoint but does include the right endpoint.

4. Why step functions matter

Step functions are helpful when the answer must be a whole number or when something is grouped into fixed units.

Common examples include:

• the number of buses needed for students
• the number of cartons needed for products
• phone plans charged by each started minute
• tickets sold only in whole numbers
• wages based on completed hours

If you cannot use part of a unit, you often need the ceiling function. If only completed units count, you often need the floor function.

5. How to tell whether to use floor or ceiling

• Use floor when you count only full or completed parts.
• Use ceiling when you need enough whole units to cover the amount.

For example:

• If a worker is paid only for full hours completed, floor may be used.
• If 26 students need buses that hold 10 each, ceiling must be used, because 2.6 buses means you really need 3 buses.

6. Reading step functions from a rule

A step function can be written using floor or ceiling symbols.

For example, consider:

$$y = \lfloor x \rfloor + 2$$

This means:

• first find \(\lfloor x \rfloor\)
• then add 2

So:

• if \(1 \le x < 2\), then \(\lfloor x \rfloor = 1\), so \(y = 3\)
• if \(2 \le x < 3\), then \(\lfloor x \rfloor = 2\), so \(y = 4\)

This shifts the basic floor graph upward by 2 units.

Another example is:

$$y = \lceil x/2 \rceil$$

This creates wider steps because the input is divided by 2 before rounding up.

7. Worked Examples

Example 1: Evaluate a floor function

Find the value of \(f(x) = \lfloor x \rfloor\) when:

• \(x = 4.7\)
• \(x = -2.3\)
• \(x = 6\)

Solution:

• \(\lfloor 4.7 \rfloor = 4\) because 4 is the greatest integer less than or equal to 4.7
• \(\lfloor -2.3 \rfloor = -3\) because -3 is the greatest integer less than or equal to -2.3
• \(\lfloor 6 \rfloor = 6\) because 6 is already an integer

Answer: \(4\), \(-3\), and \(6\)

Example 2: Evaluate a ceiling function

Find the value of \(g(x) = \lceil x \rceil\) when:

• \(x = 2.1\)
• \(x = -4.8\)
• \(x = 9\)

Solution:

• \(\lceil 2.1 \rceil = 3\)
• \(\lceil -4.8 \rceil = -4\)
• \(\lceil 9 \rceil = 9\)

Answer: \(3\), \(-4\), and \(9\)

Example 3: Real-world ceiling situation

A box can hold 12 tennis balls. How many boxes are needed for 50 tennis balls?

Solution:

Divide the number of balls by the number each box can hold:

$$50 \div 12 = 4.166\ldots$$

You cannot use 4.166 boxes. You need enough whole boxes to hold all 50 balls, so round up:

$$\left\lceil \frac{50}{12} \right\rceil = 5$$

Answer: 5 boxes

Example 4: Real-world floor situation

A video game gives 1 reward point for each full 15 coins collected. If a player collects 52 coins, how many reward points do they earn?

Solution:

We count only complete groups of 15:

$$\frac{52}{15} = 3.466\ldots$$

Only 3 full groups are completed, so use floor:

$$\left\lfloor \frac{52}{15} \right\rfloor = 3$$

Answer: 3 reward points

8. How to sketch a basic step function

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

1. Choose an interval such as \(0 \le x < 1\). On this interval, draw a horizontal line at \(y = 0\).
2. Then draw a horizontal line at \(y = 1\) for \(1 \le x < 2\).
3. Continue this pattern for other intervals.
4. Use a closed dot where the endpoint is included.
5. Use an open dot where the endpoint is not included.

For \(y = \lfloor x \rfloor\), the left endpoints are closed and the right endpoints are open.

For \(y = \lceil x \rceil\), the left endpoints are open and the right endpoints are closed.

9. Important ideas to remember

• A step function has constant pieces and sudden jumps.
• The floor function \(\lfloor x \rfloor\) rounds down.
• The ceiling function \(\lceil x \rceil\) rounds up.
• Step functions are useful in real-life situations where only whole units make sense.
• The graph shows horizontal segments instead of a smooth curve.

10. Quick check questions

• What is \(\lfloor 8.9 \rfloor\)?
• What is \(\lceil 8.9 \rceil\)?
• What is \(\lfloor -3.1 \rfloor\)?
• What is \(\lceil -3.1 \rceil\)?
• If 73 people need vans that hold 8 people each, should you use floor or ceiling?

Answers:

• \(\lfloor 8.9 \rfloor = 8\)
• \(\lceil 8.9 \rceil = 9\)
• \(\lfloor -3.1 \rfloor = -4\)
• \(\lceil -3.1 \rceil = -3\)
• Use ceiling, because you need enough vans for everyone.

Summary

Step functions model situations where outputs change in jumps instead of smoothly. The floor function \(\lfloor x \rfloor\) rounds down to the greatest integer less than or equal to \(x\), and the ceiling function \(\lceil x \rceil\) rounds up to the smallest integer greater than or equal to \(x\).

When solving real-life problems, ask yourself whether you are counting only completed groups or whether you need enough whole units. That will help you decide whether to use floor or ceiling.

Function Transformations describe how a graph changes when we adjust a function rule. Instead of drawing a brand-new graph every time, we can start with a basic function and predict how it will move, flip, or stretch.

This is very useful because many functions are built from simple ones. If you understand transformations, you can look at an expression like \(f(x-3)+2\) and quickly tell what happens to the graph.

In this lesson, you will learn how to recognize and describe translations, reflections, and dilations using function notation.

1. Start with a parent function

A parent function is the basic version of a function. Some common examples are:

• Linear: \(f(x)=x\)
• Quadratic: \(f(x)=x^2\)
• Absolute value: \(f(x)=|x|\)

Transformations change the graph of the parent function, but the general shape usually stays related to the original.

For example, if the parent function is

$$f(x)=x^2$$

then the graph is a parabola opening upward with vertex at \((0,0)\).

2. The basic transformation form

A very common transformation form is:

$$g(x)=a\,f(x-h)+k$$

Each part changes the graph in a different way:

• \(h\): horizontal shift
• \(k\): vertical shift
• \(a\): reflection and vertical stretch/shrink

Let us look at each one carefully.

3. Vertical translations: \(f(x)+k\)

When a number is added outside the function, it moves the graph up or down.

• \(f(x)+k\): move up \(k\) units if \(k>0\)
• \(f(x)-k\): move down \(k\) units

Example:

$$g(x)=f(x)+3$$

This moves the graph of \(f(x)\) up 3 units.

If a point on the original graph is \((2,5)\), then on the new graph it becomes \((2,8)\).

Key idea: Vertical changes affect the output values.

4. Horizontal translations: \(f(x-h)\)

When a number is added or subtracted inside the function, the graph moves left or right.

• \(f(x-h)\): move right \(h\) units
• \(f(x+h)\): move left \(h\) units

This often feels backward at first. The sign inside the function does the opposite of what you might expect.

Example:

$$g(x)=f(x-4)$$

This moves the graph of \(f(x)\) right 4 units.

If a point on the original graph is \((1,6)\), then on the new graph it becomes \((5,6)\).

Key idea: Horizontal changes affect the input values.

5. Reflections

A reflection flips a graph over an axis.

Reflection across the x-axis

If we multiply the whole function by \(-1\), we get:

$$g(x)=-f(x)$$

This reflects the graph across the x-axis. Every output changes sign.

For example, if \((3,2)\) is on \(f(x)\), then \((3,-2)\) is on \(-f(x)\).

Reflection across the y-axis

If we replace \(x\) with \(-x\), we get:

$$g(x)=f(-x)$$

This reflects the graph across the y-axis. Every input changes sign.

For example, if \((4,1)\) is on \(f(x)\), then \((-4,1)\) is on \(f(-x)\).

6. Vertical dilations: stretches and shrinks

When we multiply the function by a number, we change how tall the graph looks.

$$g(x)=a\,f(x)$$

• If \(|a|>1\), the graph is stretched vertically.
• If \(0<|a|<1\), the graph is shrunk vertically.
• If \(a<0\), there is also a reflection across the x-axis.

Examples:

• \(g(x)=2f(x)\): vertical stretch by factor 2
• \(g(x)=\frac{1}{2}f(x)\): vertical shrink by factor \(\frac{1}{2}\)
• \(g(x)=-3f(x)\): reflect across x-axis and stretch by factor 3

7. Reading the full form \(a\,f(x-h)+k\)

In the function

$$g(x)=a\,f(x-h)+k$$

• \(x-h\) means shift right \(h\)
• \(+k\) means shift up \(k\)
• \(a\) changes the height of the graph
• If \(a\) is negative, reflect across the x-axis

You can think of it as: inside changes horizontal position, outside changes vertical position.

8. Worked Example 1: Simple vertical and horizontal shifts

Suppose the parent function is

$$f(x)=x^2$$

Describe the transformation in

$$g(x)=f(x-2)+3$$

Step 1: Look inside the function.

The expression is \(x-2\), so the graph moves right 2.

Step 2: Look outside the function.

The \(+3\) moves the graph up 3.

Answer: The graph of \(y=x^2\) shifts right 2 units and up 3 units.

The original vertex is \((0,0)\), so the new vertex is \((2,3)\).

9. Worked Example 2: Reflection and vertical stretch

Let

$$f(x)=|x|$$

Describe the transformation in

$$g(x)=-2f(x)$$

Step 1: The negative sign outside the function reflects the graph across the x-axis.

Step 2: The factor of 2 causes a vertical stretch by 2.

Answer: The graph of \(y=|x|\) is flipped over the x-axis and stretched vertically by a factor of 2.

The point \((1,1)\) on \(f(x)\) becomes \((1,-2)\) on \(g(x)\).

10. Worked Example 3: A full transformation

Let

$$f(x)=x^2$$

Describe

$$g(x)=-f(x+1)-4$$

Step 1: \(x+1\) inside the function means shift left 1.

Step 2: The negative sign outside reflects the graph across the x-axis.

Step 3: The \(-4\) outside shifts the graph down 4.

Answer: The graph shifts left 1, reflects across the x-axis, and moves down 4.

The original vertex \((0,0)\) becomes \((-1,-4)\).

11. Worked Example 4: Using points to track a transformation

Suppose a function \(f(x)\) has a point \((2,5)\). What point will be on the graph of

$$g(x)=f(x-3)+2$$

Step 1: \(x-3\) means shift right 3.

So the x-coordinate changes from 2 to 5.

Step 2: \(+2\) means shift up 2.

So the y-coordinate changes from 5 to 7.

Answer: The new point is \((5,7)\).

12. Common mistakes to avoid

• Horizontal shifts feel backward. \(f(x-2)\) moves right, not left.
• Outside numbers affect y-values. \(+3\) outside means up 3.
• Inside numbers affect x-values. \(+3\) inside means left 3.
• A negative outside reflects across the x-axis.
• A negative inside reflects across the y-axis.

13. Quick transformation guide

• \(f(x)+k\): up \(k\)
• \(f(x)-k\): down \(k\)
• \(f(x-h)\): right \(h\)
• \(f(x+h)\): left \(h\)
• \(-f(x)\): reflect across x-axis
• \(f(-x)\): reflect across y-axis
• \(a f(x)\): vertical stretch or shrink

14. How to analyze any transformed function

1. Find the parent function.
2. Look inside the function for horizontal changes.
3. Look outside the function for vertical changes.
4. Check for a negative sign to see if there is a reflection.
5. Check for a number multiplied outside to see if there is a stretch or shrink.

15. Summary

Function transformations tell you how a graph changes from a parent function. In \(a\,f(x-h)+k\), the value of \(h\) moves the graph left or right, \(k\) moves it up or down, and \(a\) can stretch, shrink, or reflect it.

Remember the most important pattern: inside affects horizontal movement, outside affects vertical movement. With practice, you will be able to read a function rule and predict the graph without plotting every point.