16.3: Graphs of Functions

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Learning Objectives

By the end of this section, you will be able to:

Use the vertical line test
Identify graphs of basic functions
Read information from a graph of a function

Before you get started, take this readiness quiz.

Evaluate: ⓐ $2^3$ ⓑ $3^2$.
If you missed this problem, review [link].
Evaluate: ⓐ $|7|$ ⓑ $|−3|$.
If you missed this problem, review [link].
Evaluate: ⓐ $\sqrt{4}$ ⓑ $\sqrt{16}$.
If you missed this problem, review [link].

Use the Vertical Line Test

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.

An ordered pair $(x,y)$ is a solution of a linear equation, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.

The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.

In Figure, we can see that, in graph of the equation $y=2x−3$, for every x-value there is only one y-value, as shown in the accompanying table.

plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled y equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label y equals2 x minus 3. The second row is a header row with the headers x, y, and (x, y). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5). — Figure $\PageIndex{1}$

A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation $y=2x−3$ is a function.

If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every x-value there is only one y-value.

If the vertical line hit the graph twice, the x-value would be mapped to two y-values, and so the graph would not represent a function.

This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

VERTICAL LINE TEST

A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.

If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Example $\PageIndex{1}$

Determine whether each graph is the graph of a function.

Answer

ⓐ Since any vertical line intersects the graph in at most one point, the graph is the graph of a function.

$The figure has a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 5, x equalsnegative 3, and x equals3. Each line intersects the slanted line at exactly one point.$

ⓑ One of the vertical lines shown on the graph, intersects it in two points. This graph does not represent a function.

Example $\PageIndex{2}$

Determine whether each graph is the graph of a function.

$The figure has two graphs. In graph a there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (0, negative 1), (negative 1, 0), (1, 0), (negative 2, 3), and (2, 3). In graph b there is a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 2, 0), (2, 0), (0, negative 2), and (0, 2).$

Answer: ⓐ yes ⓑ no

Example $\PageIndex{3}$

Determine whether each graph is the graph of a function.

$The figure has two graphs. In graph a there is an ellipse graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The ellipse goes through the points (0, negative 3), (negative 2, 0), (2, 0), and (0, 3). In graph b there is a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (2, 0), and (4, 2).$

Answer: ⓐ no ⓑ yes

Identify Graphs of Basic Functions

We used the equation $y=2x−3$ and its graph as we developed the vertical line test. We said that the relation defined by the equation $y=2x−3$ is a function.

We can write this as in function notation as $f(x)=2x−3$. It still means the same thing. The graph of the function is the graph of all ordered pairs $(x,y)$ where $y=f(x)$. So we can write the ordered pairs as $(x,f(x))$. It looks different but the graph will be the same.

Compare the graph of $y=2x−3$ previously shown in Figure with the graph of $f(x)=2x−3$ shown in Figure. Nothing has changed but the notation.

This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5). — Figure $\PageIndex{2}$

GRAPH OF A FUNCTION

The graph of a function is the graph of all its ordered pairs, (x,y)(x,y) or using function notation, (x,f(x))(x,f(x)) where y=f(x).y=f(x).

\[\begin{array} {ll} {f} &{\text{name of function}} \\ {x} &{\text{x-coordinate of the ordered pair}} \\ {f(x)} &{\text{y-coordinate of the ordered pair}} \\ \nonumber \end{array}\]

As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.

Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if $y=2x−3$ is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the x-value has infinitely many y-values.

We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is $y=mx+b$. In function notation, this linear function becomes $f(x)=mx+b$ where m is the slope of the line and b is the y-intercept.

The domain is the set of all real numbers, and the range is also the set of all real numbers.

LINEAR FUNCTION

$This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.$

We will use the graphing techniques we used earlier, to graph the basic functions.

Example $\PageIndex{4}$

Graph: $f(x)=−2x−4$.

Answer

	$f(x)=−2x−4$
We recognize this as a linear function.
Find the slope and y-intercept.	$m=−2$ $b=−4$
Graph using the slope intercept.

Example $\PageIndex{5}$

Graph: $f(x)=−3x−1$

Answer: $The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (1, negative 4), (0, negative 1), and (negative 1, 2).$

Example $\PageIndex{6}$

Graph: $f(x)=−4x−5$

Answer

The next function whose graph we will look at is called the constant function and its equation is of the form $f(x)=b$, where b is any real number. If we replace the $f(x)$ with y, we get $y=b$. We recognize this as the horizontal line whose y-intercept is b. The graph of the function $f(x)=b$, is also the horizontal line whose y-intercept is b.

Notice that for any real number we put in the function, the function value will be b. This tells us the range has only one value, b.

CONSTANT FUNCTION

$This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.$

Example $\PageIndex{7}$

Graph: $f(x)=4$.

Answer

	$f(x)=4$
We recognize this as a constant function.
The graph will be a horizontal line through $(0,4)$.	$.$

Example $\PageIndex{8}$

Graph: $f(x)=−2$.

Answer: $The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).$

Example $\PageIndex{9}$

Graph: $f(x)=3$.

Answer: $The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3), (1, 3), and (2, 3).$

The identity function, $f(x)=x$ is a special case of the linear function. If we write it in linear function form, $f(x)=1x+0$, we see the slope is 1 and the y-intercept is 0.

IDENTITY FUNCTION

$This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.$

The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.

Graph: $f(x)=x^2$.

Answer

We choose x-values. We substitute them in and then create a chart as shown.

$This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9). The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx squared, and (x, f of x). The second row has the coordinates negative 3, 9, and (negative 3, 9). The third row has the coordinates negative 2, 4, and (negative 2, 4). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 4, and (2, 4). The seventh row has the coordinates 3, 9, and (3, 9).$

Example $\PageIndex{11}$

Graph: $f(x)=x^2$.

Answer: $This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).$

Example $\PageIndex{12}$

$f(x)=−x^2$

Answer: $This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, negative 4), (negative 1, negative 1), (0, 0), (1, negative 1), and (2, negative 4).$

Looking at the result in Example, we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.

The range is not all real numbers. Notice the graph consists of values of y never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.

SQUARE FUNCTION

$This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.$

The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our x-values.

Graph: $f(x)=x^3$.

Answer

We choose x-values. We substitute them in and then create a chart.

$This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8). Next to the graph is a table. The table has 6 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx cubed, and (x, f of x). The second row has the coordinates negative 2, negative 8, and (negative 2, negative 8). The third row has the coordinates negative 1, negative 1, and (negative 1, negative 1). The fourth row has the coordinates 0, 0, and (0, 0). The fifth row has the coordinates 1, 1, and (1, 1). The sixth row has the coordinates 2, 8, and (2, 8).$

Example $\PageIndex{14}$

Graph: $f(x)=x^3$.

Answer: $This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).$

Example $\PageIndex{15}$

Graph: $f(x)=−x^3$.

Answer

Looking at the result in Example, we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.

The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.

CUBE FUNCTION

The next function we will look at does not square or cube the input values, but rather takes the square root of those values.

Let’s graph the function $f(x)=\sqrt{x}$ and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers.

Example $\PageIndex{16}$

$f(x)=\sqrt{x}$

Answer

We choose x-values. Since we will be taking the square root, we choose numbers that are perfect squares, to make our work easier. We substitute them in and then create a chart.

$This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph is a table. The table has 5 rows and 3 columns. The first row is a header row with the headers x, f of x equalssquare root of x, and (x, f of x). The second row has the coordinates 0, 0, and (0, 0). The third row has the coordinates 1, 1, and (1, 1). The fourth row has the coordinates 4, 2, and (4, 2). The fifth row has the coordinates 9, 3, and (9, 3).$

Example $\PageIndex{17}$

Graph: $f(x)=x$.

Answer: $This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1), (4, 2), and (9, 3).$

Example $\PageIndex{18}$

Graph: $f(x)=−\sqrt{x}$.

Answer: $This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from negative 10 to 0. The curved half-line starts at the point (0, 0) and then goes down and to the right. The curved half line goes through the points (1, negative 1), (4, negative 2), and (9, negative 3).$

SQUARE ROOT FUNCTION

Our last basic function is the absolute value function, $f(x)=|x|$. Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range.

Graph: $f(x)=|x|$.

Answer

We choose x-values. We substitute them in and then create a chart.

$This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). Next to the graph is a table. The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsabsolute value of x, and (x, f of x). The second row has the coordinates negative 3, 3, and (negative 3, 3). The third row has the coordinates negative 2, 2, and (negative 2, 2). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 2, and (2, 2). The eighth row has the coordinates 3, 3, and (3, 3).$

Example $\PageIndex{20}$

Graph: $f(x)=|x|$.

Answer: $This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3).$

Example $\PageIndex{21}$

Graph: $f(x)=−|x|$.

Answer: $This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The v-shaped line goes through the points (negative 3, negative 3), (negative 2, negative 2), (negative 1, negative 1), (0, 0), (1, negative 1), (2, negative 2), and (3, negative 3).$

ABSOLUTE VALUE FUNCTION

Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.

We will start by reading the domain and range of a function from its graph.

Remember the domain is the set of all the x-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of x that have a corresponding value on the graph. Follow the value x up or down vertically. If you hit the graph of the function then x is in the domain.

Remember the range is the set of all the y-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of y that have a corresponding value on the graph. Follow the value y left or right horizontally. If you hit the graph of the function then y is in the range.

Example $\PageIndex{22}$

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

$This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line segment goes through the points (negative 3, negative 1), (1.5, 3), and (3, 1). The interval [negative 3, 3] is marked on the horizontal axis. The interval [negative 1, 3] is marked on the vertical axis.$

Answer

To find the domain we look at the graph and find all the values of x that correspond to a point on the graph. The domain is highlighted in red on the graph. The domain is $[−3,3]$.

To find the range we look at the graph and find all the values of y that correspond to a point on the graph. The range is highlighted in blue on the graph. The range is $[−1,3]$.

Example $\PageIndex{23}$

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

$This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 5, negative 4), (0, negative 3), and (1, 2). The interval [negative 5, 1] is marked on the horizontal axis. The interval [negative 4, 2] is marked on the vertical axis.$

Answer: The domain is $[−5,1]$. The range is $[−4,2]$.

Example $\PageIndex{24}$

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

$This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 5. The y-axis runs from negative 6 to 4. The curved line segment goes through the points (negative 2, 1), (0, 3), and (4, negative 5). The interval [negative 2, 4] is marked on the horizontal axis. The interval [negative 5, 3] is marked on the vertical axis.$

Answer: The domain is $[−2,4]$. The range is $[−5,3]$.

Access this online resource for additional instruction and practice with graphs of functions.

Find Domain and Range

Key Concepts

Vertical Line Test
- A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
- If any vertical line intersects the graph in more than one point, the graph does not represent a function.
Graph of a Function
- The graph of a function is the graph of all its ordered pairs, (x,y)(x,y) or using function notation, (x,f(x))(x,f(x)) where y=f(x).y=f(x).
  fxf(x)name of functionx-coordinate of the ordered pairy-coordinate of the ordered pairfname of functionxx-coordinate of the ordered pairf(x)y-coordinate of the ordered pair
Linear Function
$This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.$
Constant Function
$This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.$
Identity Function
$This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.$
Square Function
$This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.$
Cube Function
Square Root Function
Absolute Value Function