Finding Domain and Range from Graphs

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Course:	MA120: Applied College Algebra
Book:	Finding Domain and Range from Graphs

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Finding Domain and Range from Graphs
Finding Domains and Ranges of the Toolkit Functions
Find the Domain of a Function From a Graph

Finding Domain and Range from Graphs

In this section, we will find the domain and range of functions given their graphs.

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8.

Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range

Figure 8

We can observe that the graph extends horizontally from $-5$ to the right without bound, so the domain is $[-5, \infty)$ . The vertical extent of the graph is all range values $5$ and below, so the range is $(-\infty, 5]$ . Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Example 6

Finding Domain and Range from a Graph

Find the domain and range of the function $f$ whose graph is shown in Figure 9.

Graph of a function from (-3, 1].

Figure 9

Solution

We can observe that the horizontal extent of the graph is $-3$ to $1$ , so the domain of $f$ is $(-3,1]$ .

The vertical extent of the graph is $0$ to $-4$ , so the range is $[-4,0)$ . See Figure 10.

Graph of the previous function shows the domain and range.

Figure 10

Example 7

Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function $f$ whose graph is shown in Figure 11.

Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.

Figure 11

Solution

The input quantity along the horizontal axis is "years," which we represent with the variable $t$ for time. The output quantity is "thousands of barrels of oil per day," which we represent with the variable $b$ for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as $1973 \leq t \leq 2008$ and the range as approximately $180 \leq b \leq 2010$ .

In interval notation, the domain is $[1973,2008]$ , and the range is about $[180,2010]$ . For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Try It #6

Given Figure 12, identify the domain and range using interval notation.

Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.

Figure 12

Q&A

Can a function's domain and range be the same?

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Source: Rice University, https://openstax.org/books/college-algebra/pages/3-2-domain-and-range
This work is licensed under a Creative Commons Attribution 4.0 License.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Constant function f(x)=c.

Figure 13 For the constant function $f(x)=c$ , the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant $c$ , so the range is the set $\{c\}$ that contains this single element. In interval notation, this is written as $[c, c]$ , the interval that both begins and ends with $c$ .

Identity function f(x)=x.

Figure 14 For the identity function $f(x)=x$ , there is no restriction on $x$ . Both the domain and range are the set of all real numbers.

Absolute function f(x)=|x|.

Figure 15 For the absolute value function $f(x)=|x|$ , there is no restriction on $x$ . However, because absolute value is defined as a distance from $0$ , the output can only be greater than or equal to $0$ .

Quadratic function f(x)=x^2.

Figure 16 For the quadratic function $f(x)=x^{2}$ , the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

Cubic function f(x)-x^3.

Figure 17 For the cubic function $f(x)=x^{3}$ , the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.

Reciprocal function f(x)=1/x.

Figure 18 For the reciprocal function $f(x)=\frac{1}{x}$ , we cannot divide by $0$ , so we must exclude $0$ from the domain. Further, 1 divided by any value can never be $0$ , so the range also will not include 0. In set-builder notation, we could also write $\{x \mid x \neq 0\}$ , the set of all real numbers that are not zero.

Reciprocal squared function f(x)=1/x^2

Figure 19 For the reciprocal squared function $f(x)=\frac{1}{x^{2}}$ , we cannot divide by 0, so we must exclude $0$ from the domain. There is also no $x$ that can give an output of $0$ , so $0$ is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.

Square root function f(x)=sqrt(x).

Figure 20 For the square root function $f(x)=\sqrt{x}$ , we cannot take the square root of a negative real number, so the domain must be $0$ or greater. The range also excludes negative numbers because the square root of a positive number $x$ is defined to be positive, even though the square of the negative number $-\sqrt{x}$ also gives us $x$ .

Cube root function f(x)=x^(1/3).

Figure 21 For the cube root function $f(x)=\sqrt[3]{x}$ , the domain and range include all real numbers. Note that there is no problem taking a cube root, or any oddinteger root, of a negative number, and the resulting output is negative (it is an odd function).

HOW TO

Given the formula for a function, determine the domain and range.

Exclude from the domain any input values that result in division by zero.
Exclude from the domain any input values that have nonreal (or undefined) number outputs.
Use the valid input values to determine the range of the output values.
Look at the function graph and table values to confirm the actual function behavior.

Example 8

Finding the Domain and Range Using Toolkit Functions

Find the domain and range of $f(x)=2 x^{3}-x$ .

Solution

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is $(-\infty, \infty)$ and the range is also $(-\infty, \infty)$ .

Example 9

Finding the Domain and Range

Find the domain and range of $f(x)=\frac{2}{x+1}$ .

Solution

We cannot evaluate the function at $-1$ because division by zero is undefined. The domain is $(-\infty,-1) \cup(-1, \infty)$ . Because the function is never zero, we exclude $0$ from the range. The range is $(-\infty, 0) \cup(0, \infty)$ .

Example 10

Finding the Domain and Range

Find the domain and range of $f(x)=2 \sqrt{x+4}$ .

Solution

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

$x+4 \geq 0 \text { when } x \geq-4$

The domain of $f(x)$ is $[-4, \infty)$ .

We then find the range. We know that $f(-4)=0$ , and the function value increases as $x$ increases without any upper limit. We conclude that the range of $f$ is $[0, \infty)$ .