1.2 Functions

To Learning Outcomes

Determine if a graph is the graph of a function by using the Vertical Line Test.
Find the value of a function from a graph.
State the domain and range from the graph of a function in set-builder, inequality, and/or interval notation.
Evaluate functions at a given input value using an equation. Also, determine inputs resulting in the given output value
Using the equation of a function, state the domain in set-builder, inequality, and/or interval notation.

Functions

You should be familiar with functions from previous math courses. Here, we’ll do a quick review. Some people think of functions as “mathematical machines.” Imagine you have a machine that changes a number according to a specific rule such as “multiply by $3$ and add $2$ ” or “divide by $5$ , add $25$ , and multiply by $−1$ .” If you put a number into the machine, a new number will come out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.

You can also call the machine “ $f$ ” for function. If you put $x$ into the machine, $f(x)$ (read “ $f$ of $x$ “), comes out. Mathematically speaking, $x$ is the input, or the independent variable, and $f(x)$ is the output, or the dependent variable, since it depends on the value of $x$ .

Relations and Functions

In mathematics, a relation is a set of ordered pairs. The set of first components of the ordered pairs is called the domain and the set of second components of the ordered pairs is called the range. In a set of ordered pairs, you find the domain by listing all of the input values (the $x$ -coordinates). To find the range, list all of the output values (the $y$ -coordinates).

A function is a special type of relation in which each domain value corresponds to exactly one range value.

If you find that you need more help understanding functions, please click on the link to learn more about functions.

Consider the following relation: $\{(−2,0),(0,6),(-2,12),(4,18)\}$ .

For this relation, the domain is $\{−2,0,4\}$ and the range is $\{0,6,12,18\}$ .

Is the above relation a function or not?

This relation is NOT a function because the input values are not unique, in other words, there is a repeated $x$ -value that corresponds to different $y$ -values.

Example

List the domain and range for the following relation where $x$ is the input and $f(x)$ is the output and determine whether the relation given is a function. Remember each pair of $x$ and $f(x)$ exists as a point $(x,f(x))$ on the graph of the function .

$x$	$f(x)$
$−3$	$4$
$−2$	$2$
$−1$	$1$
$2$	$4$
$3$	$4$

Show Solution

The domain describes all the inputs, and we can use set notation with brackets { } to make the list. The domain is $\{-3,-2,-1,2,3\}$ . The range describes all the outputs. The range is $\{1,2,4\}$ . Notice that $4$ is only listed once in the list of elements in the range.

This relation is a function because each unique input (no repeated $x$ values) has a single output. Notice also that $4$ is a repeated $f(x)$ -value but that has no effect on whether it is a function or not.

In the following video we provide another example of identifying domain and range. Determine whether the table of values represents a function.

In the above video, each domain value corresponds to exactly one range value. No $x$ -coordinates were repeated so that relation is a function.

Function Notation

The notation $y=f\left(x\right)$ defines a function named $f$ . The letter $x$ represents the input value, or independent variable. The letter $y$ or $f\left(x\right)$ , represents the output value, or dependent variable. When a function is defined by an equation, we usually use function notation.

For example, the equation $y=3x+1$ may also be written as $f(x)=3x+1$ . $f$ is the name of the function, $x$ is an input from the domain of the function, and $f(x)$ is the function value at $x$ also called the output.

Inputs of Functions

A function can be evaluated at different values of $x$ . Given the function $f(x)=3x+1$ , find $f(-2)$ . In other words, if $x$ is $-2$ find $f(x)$ . We must substitute $\color{Green}{-2}$ in place of $\color{Green}{x}$ and evaluate.
$\begin{align} f(\color{Green}{-2}\color{black}{)} &= 3(\color{Green}{-2}\color{black}{)+1}\\ &=-6+1\\ &= -5 \end{align}$

So, when $x=-2$ , $f(x)=-5$ (or $y=-5$ ). This can be written as the ordered pair $(-2,-5)$ and it is a point on the graph of the line $f(x)=3x+1$ .

It is important to note that the parentheses that are part of function notation do not mean multiply. The notation $f(x)$ does not mean $f$ multiplied by $x$ . Instead, the notation means “ $f\text{ of } x$ ” or “the function of $x$ .” To evaluate the function, take the value given for $x$ , and substitute that value in for $x$ in the expression.

Let us look at a couple of examples.

ExAMple

Given $f(x)=3x–4$ , evaluate $f(5)$ .

Show Solution

Substitute $\color{Green}{5}$ in for $\color{Green}{x}$ in the function. Then simplify the expression on the right side of the equation.

$\begin{align} f(\color{Green}{5}) &= 3(\color{Green}{5})-4\\ &= 15-4\\ &= 11 \end{align}$

We were given $x=5$ and evaluated and found that $f(5)=11$ , which is the point $(5,11)$ on the graph of the function $f(x)=3x-4$ .

Example

Given $p(x)=-2x^{2}+5$ , evaluate $p(−3)$ .

Show Solution

Substitute $\color{green}{-3}$ in for $x$ in the function.

$\begin{align} p(\color{Green}{−3}) &= -2(\color{Green}{−3})^{2}+5\\ &= -2(9)+5\\ &=-18+5\\ &= -13\end{align}$

This is the point $(-3,-13)$ on the graph of the function $f(x)=-2x^2+5$ .

Example

Given $f(x)=|4x-3|$ , evaluate $f(0)$ , $f(2)$ , and $f(−1)$ .

Show Solution

Treat these like three separate problems. In each case, you substitute the value in for $x$ and simplify.

Evaluate $f(0)$ :

Start with $\color{green}{x=0}$ and evaluate $f(x)$ for $\color{green}{x=0}$ .

$\begin{align} f(\color{green}{0}) &= |4(\color{green}{0})-3|\\ &= |0-3|\\ &= |-3|\\ &= 3\end{align}$

$f(0)=3$

Evaluate $f(2)$ :

Evaluate $f(x)$ for $\color{green}{x=2}$ .

$\begin{align}f(\color{green}{2}) &= |4(\color{green}{2})-3|\\ &= |8-3|\\ &= |5|\\ &= 5 \end{align}$

$f(2)=5$

Evaluate $f(-1)$ :

Evaluate $f(x)$ for $\color{green}{x=−1}$ .

$\begin{align} f(\color{green}{−1}) &= |4(\color{green}{-1})-3|\\ &= |-4-3|\\ &= |-7|\\ &= 7 \end{align}$

$f(-1)=7$

Outputs of Functions

Reconsider the equation $y=4x+1$ , which represents a function. If instead of having a given input value, you have a known output value, you can find the input value that would give that output. For example, for an output value of $-7$ , you can find the input value. First substitute $\color{Green}{y=-7}$ into the equation and then solve for $x$ .

$\begin{align}\color{Green}{y} &= 4x+1\\ \color{Green}{-7} &= 4x+1\\ -7-1 &= 4x\\ -8 &= 4x\\ \frac{-8}{4} &= x\\ -2 &= x\\ x &= -2\end{align}$

The output of $-7$ corresponds to an input of $-2$ . These values make an ordered pair, $(-2, -7)$ , that represents a point on the graph of this function. We will discuss this idea more fully later in this section.

In the next example, you will not be given the $x$ -value but instead you will be given a $y$ -value and asked to find the $x$ -value.

Example

Given $f(x)=3x–4$ , find the $x$ -value that yields an output of $5$ .

Show Solution

First write the equation with $x$ and $y$

$y=3x-4$

Then substitute $\color{green}{5}$ in for $y$ in the equation.

$\color{green}{5}=3x-4$

Solve for $x$ .

$\begin{align}5&=3x-4\\5+4&=3x\\9&=3x\\\frac{9}{3}&=x\\3&=x\\x&=3\\\end{align}$

For an output of $5$ , the input is $3$ .

These values make an ordered pair, $(3, 5)$ , that represents a point on the graph of this function.

Try It

Graphs of Functions and the Vertical Line Test

When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the $x$ -axis and the dependent value is plotted on the $y$ -axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as $y = f(x)$ , or “ $y$ is a function of $x$ “, we would write ordered pairs $(x, f(x))$ using function notation instead of $(x,y)$ as you may have seen previously.

We can identify whether the graph of a relation represents a function because for each $x$ -coordinate there will be exactly one $y$ -coordinate. The blue relation graphed below represents a function. Notice that when a vertical line is placed over the graph, the line does not intersect the graph more than once for any value of $x$ . This indicates that the relation is a function. A graph of a semicircle. Four vertical lines cross the semicircle at one point each. If instead a graph shows two or more intersections with a vertical line, then an input ( $x$ -coordinate) has more than one output ( $y$ -coordinate), and $y$ is not a function of $x$ .

Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the Vertical Line Test.

Example

Use the vertical line test to determine whether the relation plotted on this graph is a function.

Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)

Show Solution

This relationship cannot be a function, because some of the $x$ -coordinates have two corresponding $y$ -coordinates.

A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.

Find the Value of a Function from a Graph

Example

A function is graphed below.

graph of y=x

For the given function, evaluate $f(2)$ .
For the given graph of a function, find $x$ when $f(x)=-2$ .

Show Answer

To evaluate $f(2)$ , go to $x=2$ on the graph, then vertically to the graph of the function. The $y$ -value at this point is $2$ . Then $f(2)=2$ .
To find $x$ when $f(x)=-2$ using the graph, go to $y=-2$ , then horizontally to the graph of the function. The $x$ -value at this point is $-2$ . Then $f(x)=-2$ when $x=-2$ .

ExAmple

A function is graphed below.

y=absolute value of x

For the given function, evaluate $f(-1)$ .
For the given function, evaluate $f(0)$ .
For the given graph of a function, find $x$ when $f(x)=2$ .

Show Answer

On the graph, go to $x=-1$ , then vertically to the graph of the function. The $y$ -value at this point is $1$ . Then $f(-1)=1$ .
On the graph, go to $x=0$ , then vertically to the graph of the function. The $y$ -value at this point is $0$ . Then $f(0)=0$ .
On the graph, go to $y=2$ , then horizontally to the graph of the function. Notice that there are two $x$ -values that have this associated $y$ -value: $x=-2$ and $x=2$ . Then $f(x)=2$ when $x=-2$ or $x=2$ .

Try It

Finding the Domain and Range of a Function from its Graph

Finding domain of different functions is often a matter of asking yourself, “what values of $x$ can this function not have?” Pictures make it easier to visualize what the domain and range are, so we will first show how to define the domain and range of functions given their graphs. What are the domain and range of the function $f(x)=x+3$ ?
This is a linear function. Remember that linear functions are lines that continue forever in each direction.

Graph of the function [latex]f(x)=x+3[/latex]

Graph of the function $f(x)=x+3$

For this function, any real number can be substituted for $x$ and you will get a meaningful output. Also, for any real number, you can always find an $x$ value that gives you that number for the output. Unless a linear function is a constant, such as $f(x)=2$ , there is no restriction on the range. For the function $f(x)=x+3$ , the domain and range are all real numbers. We can write this domain and range in a variety of formats.

	Domain	Range
Set-Builder Notation	$\{x\|\text{all real numbers}\}$	$\{y\|\text{all real numbers}\}$
Inequality Notation	[latex]-\infty	[latex]-\infty
Interval Notation	$(-\infty,\infty)$	$(-\infty,\infty)$

For the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.

Example

Use the graph to determine the domain and range of the function $f(x)=−3x^{2}+6x+1.$

Downward-opening parabola with vertex of 1, 4.

Show Solution

Examining the graph, the arrows indicate that the function continues for larger and for smaller values of $x$ . The domain is all real numbers.

The vertex, or high point of this graph, is at $(1,4)$ . From the graph, you can see that all $y$ -values of the function are less than or equal to $4$ . In other words, $f(x)\leq4$ . The range of this function is all $y$ values less than or equal to $4$ .

	set-builder notation	inequality notation	interval notation
domain	$\{x\|\text{all real numbers}\}$	[latex]-\infty	$(-\infty,\infty)$
range	$\{y\|y\le4\}$	$y\le4$	$(-\infty,4]$

Exercises

Use the graph to determine the domain and range of the real-valued function $f(x)=-2+\sqrt{x+5}.$

Radical function stemming from negative 5, negative 2.

Show Solution

From the graph, we see that the smallest $x$ -value is $x=-5$ . Then the domain is all real numbers greater than or equal to $-5$ .

The smallest $y$ -value on the graph of the function is $y=-2$ . The range will be all real numbers greater than or equal to $-2$ .

	set-builder notation	inequality notation	interval notation
domain	$\{x\|x\geq−5\}$	$x\geq−5$	$[-5,\infty)$
range	$\{y\|y\geq−2\}$	$y\geq−2$	$[-2,\infty)$

Try the following problem, remembering to look at the graph for the lowest and highest $x$ -values that appear in the graph to find the domain, then for the smallest and highest $y$ -values to find the range. Make sure to notice if the graph has arrows or endpoints. That will change how you write your answer.

Try It

In the following video we show how to define the domain and range of functions from their graphs.

Finding the Domain and Range from the Equation of a Function

For this discussion, we will restrict all functions to real number values for the domain and range. Even with this starting point, it does not mean that for any function we discuss that all real numbers can be used for $x$ . It also does not mean that all real numbers can be function output values, $f(x)$ . There may be restrictions on the domain and range. The restrictions partly depend on the type of function.

Domain restrictions

In our course, there are two main reasons why domains are restricted.

You cannot divide by $0$ .
If you have a function like $f(x)=\frac{1}{x-2}$ , the domain has to be restricted so that no inputs make the denominator zero. For this example, $x$ cannot be $2$ because if you replace $x$ with $2$ , you will end up with the denominator as zero which is undefined. The domain would be $(-\infty,2)\cup(2,\infty)$ .
You cannot take the square (or other even) root of a negative number, as the result will not be a real number.
If you have a function like $f(x)=\sqrt{x}$ , the domain has to be restricted so that we don’t take the square root of a negative number. Recall that the number or expression written under a root symbol is called the radicand. The radicand of a square (or other even) root must not be negative. Then the domain for this example would be $x\ge0$ or $[0,\infty)$ .

Example

State the domain of the function $f(x)=\dfrac{x+1}{x-3}$ .

Show Answer

We have to restrict the domain to be sure that we do not divide by zero. We need to exclude from the domain any values of $x$ that would make the denominator zero. To find those values, set the denominator equal to zero and solve.

$\begin{align}x-3&=0\\x&=3\end{align}$

Then the domain is all real values of $x$ except $3$ . This is all real numbers where $x<3$ or $x>3$ . Use the union symbol, $\cup$ , to write the two sets in the domain. In interval notation, the domain is $(-\infty,3)\cup(3,\infty)$ .

Notice that we do not need to worry about the numerator of the function being zero. Zero divided by any nonzero number is just zero, not undefined.

Example

State the domain of the function $f(x)=\sqrt{2x+1}$ .

Show Answer

When we take the square root of a negative number, the result will be a non-real number. The radicand of a square root must not be negative. The domain is all the $x$ -values that would make the radicand non-negative values. In other words, all the $x$ -values that make the radicand greater than or equal to zero. So we need to solve the inequality $2x+1\ge0$ .

$\begin{align}2x+1&\ge0\\2x&\ge0-1\\2x&\ge-1\\x&\ge-\frac{1}{2}\end{align}$

Then the domain is all real values of $x$ that are greater than or equal to $-\dfrac{1}{2}$ . Using interval notation, this domain is $[-\dfrac{1}{2},\infty)$ .

In the following video, only watch the first example given. We will talk about the other two examples later in the course.

Try It

State the domain of the function $f\left(x\right)=\sqrt{7-x}$ .

Show Solution

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for $x$ .

$\begin{align}7-x &\ge 0\\-x&\ge -7 && \color{blue}{\textsf{remember to reverse the inequality symbol when dividing or multiplying by a negative number}}\\x&\le7\end{align}$

Exclude any number greater than $7$ from the domain as those inputs would make the radicand negative. The answers are all real numbers less than or equal to $7$ , or $\left(-\infty ,7\right]$ .

Module 1: Graphs, Functions, and Linear Equations

To Learning Outcomes

Functions

Relations and Functions

Example

Function Notation

Inputs of Functions

ExAMple

Example

Example

Outputs of Functions

Example

Try It

Graphs of Functions and the Vertical Line Test

Example

Find the Value of a Function from a Graph

Example

ExAmple

Try It

Finding the Domain and Range of a Function from its Graph

Example

Exercises

Try It

Finding the Domain and Range from the Equation of a Function

Domain restrictions

Example

Example

Try It

Candela Citations