1.6: Domain and Range

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Learning Objectives

Find the domain of a function defined by an equation.
Graph piecewise-defined functions.

If you are in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time: I Am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. Figure $\PageIndex{1}$ shows the amount, in dollars, each of those movies grossed when they were released, as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. We can identify different independent and dependent variables, analyze the data, determine the domain and range, and create a function which allows us to make predictions. In this section, we will investigate methods for determining the domain and range of functions such as these.

[Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year — Figure $\PageIndex{1}$: Graph of the Top-Five Grossing Horror Movies for years 2000-2003, and a Graph of the Market Share of Horror Movies by Year. Based on data compiled by www.the-numbers.com.

Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. When determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as ticket sales and the year in the horror movie example above. We also need to consider what is mathematically permitted. For example, if we had a square root (or any even root) function, we cannot include any input resu ing in a negative number under the radical if the domain and range consist of real numbers. Or in a function expressed as a fraction, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products (Figure $\PageIndex{2}$).

alt — Figure $\PageIndex{2}$: Diagram of how a function relates two relations

We begin by finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

For example, if we were to find the domain of the function \[f(x)=4+x-x^3 \nonumber\] we would notice that there are no radicals and there is no denominator in the function so we can ask ourselves what kinds of numbers work in this equation. Any number can be cubed so we know the domain is all real numbers.

As another algebraic example, find the domain of the function: \[f(x)=\sqrt{8+x} \nonumber .\]

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.

$\begin{align*} 8+x&≥0 \\[4pt] x&≥−8\\[4pt] \end{align*}$

Now, we will include all numbers greater than or equal to $-8$ and exclude any number less than $-8$. The answers are all real numbers greater than or equal to $-8$, or $\left(-8,\infty\right)$.

As a physical example, consider the amount of time it takes to do your homework assignment. First, the least amount of time you could spend is $0$ minutes. You could not have a negative value. You also do not spend an infinite amount of time. Ideally, for each lesson, you should spend about one hour on homework. We could say the domain is between $0$ and $60$ minutes.

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $$100$ to spend, he or she would need to express the interval that is more than $0$ and less than or equal to $100$ and write $\left(0, 100\right]$. We will discuss interval notation in greater detail later.

Let us review the conventions of interval notation:

The smallest term from the interval is written first.
The largest term in the interval is written second, following a comma.
Parentheses, $($ or $)$, are used to signify that an endpoint is not included, called exclusive.
Brackets, $[$ or $]$, are used to indicate that an endpoint is included, called inclusive.

See Figure $\PageIndex{3}$ for a summary of interval notation.

alt — Figure $\PageIndex{3}$: Summary of interval notation.

Exercise $\PageIndex{1}$

a. Find the domain of the function: $f(x)=5−x+x^3 \nonumber$

Answer a: $(−\infty,\infty)$

b. Find the domain of the function: \[f(x)=\sqrt{7-x} \nonumber .\]

Answer b

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.

\[ \begin{align*} 7−x&≥0 \\[4pt] −x&≥−7\\[4pt] x&≤7 \end{align*}\]

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7, or $\left(−\infty,7\right]$.

c. Find the domain of the function \[f(x)=\sqrt{5+2x}. \nonumber\]

Answer c: \[\left[−2.5,\infty\right) \nonumber\]

d. Find the domain of the function $f(x)=\dfrac{x+1}{2−x}$.

Answer d

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for x.

\[ \begin{align*} 2−x=0 \\[4pt] −x &=−2 \\[4pt] x&=2 \end{align*}\]

Now, we will exclude 2 from the domain. The answers are all real numbers where $x<2$ or $x>2$. We can use a symbol known as the union, $\cup$,to combine the two sets. In interval notation, we write the solution:$(−\infty,2)∪(2,\infty)$.

$[Line graph of f(x).]$

In interval form, the domain of f is $(−\infty,2)\cup(2,\infty)$.

e. Find the domain of the function: \[f(x)=\dfrac{1+4x}{2x−1} \nonumber\]

Answer e: \[(−\infty,\dfrac{1}{2})\cup(\dfrac{1}{2},\infty) \nonumber\]

Using Notations to Specify Domain and Range

In the previous examples, we used intervals and lists to describe the domain of functions. We can also use inequalities or other statements that might define sets of values or data to describe the behavior of the variable in set-builder notation. For example, $\{x|10≤x<30\}$ describes the behavior of x in set-builder notation. The braces $\{\}$ are read as “the set of,” and the vertical bar $|$ is read as “such that,” so we would read$ \{x|10≤x<30\}$ as “the set of x-values such that 10 is less than or equal to x, and x is less than 30.”

Figure $\PageIndex{4}$ compares inequality notation, set-builder notation, and interval notation.

Figure $\PageIndex{4}$: Summary of notations for inequalities, set-builder, and interval.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, $\cup$,to combine two unconnected intervals. For example, the union of the sets$\{2,3,5\}$ and $\{4,6\}$ is the set $\{2,3,4,5,6\}$. It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

\[\{x| |x|≥3\}=\left(−\infty,−3\right]\cup\left[3,\infty\right)\]

Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form$\{x|\text{ statement about x}\}$ which is read as, “the set of all x such that the statement about x is true.” For example,

\[\{x|4<x≤12\} \nonumber\]

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

\[\left(4,12\right] \nonumber\]

Example $\PageIndex{1}$: Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure $\PageIndex{5}$ using inequality notation, set-builder notation, and interval notation.

[Line graph of $1<=x<=3$ and $5$] — Figure $\PageIndex{5}$: Line graph of $1≤x≤3 \text{ or } x>5$.

Solution

To describe the values, $x$, included in the intervals shown, we would say, “$x$ is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

Inequality

\[1≤x≤3 \text{ or }x>5 \nonumber\]

Set-builder Notation

\[\{x|1≤x≤3 \text{ or } x>5\}\nonumber\]

Interval notation

\[[1,3]\cup(5,\infty)\nonumber\]

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Exercise $\PageIndex{2}$

Given Figure $\PageIndex{6}$, specify the graphed set in

words
set-builder notation
interval notation

[Line graph of -2<=x, -1<=x<3.] — Figure $\PageIndex{6}$: Line graph of $-2{\leq}x$, $-1{\leq}x<3$, $-1{\leq}x<3$.

Answer a: Values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
Answer b: $\{x|x≤−2 or −1≤x<3\}$
Answer c: $\left(−∞,−2\right]\cup\left[−1,3\right)$

Finding Domain and Range from 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 $\PageIndex{7}$.

[Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range] — Figure $\PageIndex{7}$: Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range

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

Example $\PageIndex{2}$: Finding Domain and Range from a Graph

Find the domain and range of the function f whose graph is shown in Figure 1.2.8.

[Graph of a function from (-3, 1].] — Figure $\PageIndex{8}$: Graph of a function from $(-3, 1]$.

Solution

We can observe that the horizontal extent of the graph is $–3$ to $1$, so the domain of f is $\left(−3,1\right]$.

The vertical extent of the graph is $0$ to $–4$, so the range is $\left[−4,0\right]$. See Figure $\PageIndex{9}$.

You may be wondering why we included $0$ in the range. Notice that the point $(0,0)$ is included in our graph.

$corrected picture.png$ Figure $\PageIndex{9}$: Graph of the previous function shows the domain and range

Example $\PageIndex{3}$: 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 $\PageIndex{10}$.

[Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.] — Figure $\PageIndex{10}$: Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years(credit: modification of work by the U.S. Energy Information Administration)

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 $1973≤t≤2008$ and the range as approximately $180≤b≤2010$.

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.

Exercise $\PageIndex{3}$

a. Find the domain and range of $f(x)=2x^3−x$.

Answer a

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 $(−\infty,\infty)$.

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

Answer b: 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 $(−\infty,0)\cup(0,\infty)$.

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

Answer c

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

$x+4≥0$ when $x≥−4$

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 $\left[0,\infty\right)$.

Analysis

Figure $\PageIndex{11}$ represents the function $f$.

<div data-mt-source="1"><img alt="" data-cke-saved-src="http://mathwiki.ucdavis.edu/@api/deki/files/892/CNX_Precalc_Figure_01_02_020.jpg" src="http://mathwiki.ucdavis.edu/@api/deki/files/892/CNX_Precalc_Figure_01_02_020.jpg" height="237" width="350"></div> — Figure $\PageIndex{11}$: Graph of a square root function at $(-4, 0)$.

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

Answer d

domain: $\left(−\infty,-2\right]$

range: $\left[0,\infty\right)$

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function $f(x)=|x|$. With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

\[ f(x)=x \; \text{ if } \; x≥0 \nonumber \]

If we input a negative value, the output is the opposite of the input.

\[ f(x) = -x \; \text { if } \; x < 0 \nonumber \]

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S would be $0.1S$ if $S≤$10,000$ and $$1000+0.2(S−$10,000)$ if $S>$10,000$.

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

\[f(x)= \begin{cases} \text{formula 1} & \text{if x is in domain 1} \\ \text{formula 2} &\text{if x is in domain 2} \\ \text{formula 3} &\text{if x is in domain 3}\end{cases} \nonumber \]

In piecewise notation, the absolute value function is

\[|x|= \begin{cases} x & \text{if $x \geq 0$} \\ -x &\text{if $x<0$} \end{cases} \nonumber \]

Example $\PageIndex{4}$: Graphing a Piecewise Function

Sketch a graph of the function.

\[f(x)= \begin{cases} x^2 & \text{if $x \leq 1$} \\ 3 &\text{if $1<x\leq2$} \\ x &\text{if $x>2$} \end{cases} \nonumber \]

Solution

Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Figure $\PageIndex{12}$ shows the three components of the piecewise function graphed on separate coordinate systems.

$[Graph of each part of the piece-wise function f(x)]$
Figure $\PageIndex{12}$: Graph of each part of the piece-wise function f(x)

(a)$ f(x)=x^2$ if $x≤1$; (b) $f(x)=3$ if $1< x≤2$; (c) $f(x)=x$ if $x>2$

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure $\PageIndex{13}$.

[Graph of the entire function.] — Figure $\PageIndex{13}$: Graph of the entire function.

Analysis

Note that the graph does pass the vertical line test even at $x=1$ and $x=2$ because the points $(1,3)$ and $(2,2)$ are not part of the graph of the function, though $(1,1)$ and $(2, 3)$ are.

Example $\PageIndex{5}$

Graph the following piecewise function.

\[f(x)= \begin{cases} x^3 & \text{if $x < -1$} \\ -2 &\text{if $-1<x<4$} \\ \sqrt{x} &\text{if $x>4$} \end{cases} \nonumber \]

Answer: $[Graph of f(x).]$
Figure $\PageIndex{14}$

Review

The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
The domain of a function can be determined by listing the input values of a set of ordered pairs.
The domain of a function can also be determined by identifying the input values of a function written as an equation.
Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.
For many functions, the domain and range can be determined from a graph.
An understanding of toolkit functions can be used to find the domain and range of related functions.
A piecewise function is described by more than one formula.
A piecewise function can be graphed using each algebraic formula on its assigned subdomain.

Footnotes

1 The Numbers: Where Data and the Movie Business Meet. “Box Office History for Horror Movies.” http://www.the-numbers.com/market/genre/Horror. Accessed 3/24/2014
2 www.eia.gov/dnav/pet/hist/Lea...s=MCRFPAK2&f=A.

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