Based on data compiled by www.the-numbers.com.^[1]

Finding the Domain of a Function Defined by an Equation

In Section 1.1, 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. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, 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. See Figure 2.

Figure 2

For functions of real numbers, we can write the domain and range in interval notation, which uses values within brackets to describe a set of real 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 [latex]\left(0,\text{ }100\right].[/latex] We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, 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,  exclude values that would make the radicand negative.

Before we begin, let us discuss the conventions of interval notation:

• The smallest value from the interval is written first, followed by a comma.
• The largest value in the interval is written second.
• 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.
• Use a symbol known as the union, [latex]\cup ,[/latex] to combine intervals if there are more than one.

See Figure 3 for a summary of interval notation.

Figure 3

Example 3:  Finding the Domain of a Function Involving a Denominator

Find the domain of the function [latex]f\left(x\right)=\frac{x+1}{2-x}.[/latex]

Show Solution

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 [latex]x.[/latex]

[latex]\begin{align*}2-x&=0\\-x&=-2\\x&=2\end{align*}[/latex]

Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[/latex] or [latex]x>2.[/latex] We can use a symbol known as the union, [latex]\cup,[/latex] to combine the two sets. In interval notation, we write the solution: [latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right).[/latex]

Figure 4

In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,2\right)\cup \left(2,\infty \right).[/latex]

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities 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, [latex]\left\{x|10\le x<30\right\}[/latex] describes the behavior of [latex]x[/latex] in set-builder notation. The braces [latex]\left\{\right\}[/latex] are read as “the set of,” and the vertical bar | is read as “such that,” so we would read [latex]\left\{x|10\le x<30\right\}[/latex] as “the set of x-values such that 10 is less than or equal to [latex]x,[/latex] and [latex]x[/latex] is less than 30.”

Figure 5 compares inequality notation, set-builder notation, and interval notation.

Figure 5

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, [latex]\cup ,[/latex] to combine two unconnected intervals. For example, the union of the sets [latex]\left\{2,3,5\right\}[/latex] and [latex]\left\{4,6\right\}[/latex] is the set [latex]\left\{2,3,4,5,6\right\}.[/latex] 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

Example 5:  Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.

Figure 6

Show Solution

To describe the values, [latex]x,[/latex] included in the intervals shown, we would say, “[latex]x[/latex] 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	[latex]1\le x\le 3\text{ }\text{or}\text{ }x>5[/latex]
Set-builder notation	[latex]\left\{x|1\le x\le 3\text{ }\text{or}\text{ }x>5\right\}[/latex]
Interval notation	[latex]\left[1,3\right]\cup \left(5,\infty \right)[/latex]

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.

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. This is indicated by the arrows in Figure 8.

Figure 8

We can observe that the graph extends horizontally from [latex]-5[/latex] to the right without bound, so the domain is [latex]\left[-5,\infty \right).[/latex] The vertical extent of the graph is all range values [latex]5[/latex] and below, so the range is [latex]\left(\mathrm{-\infty },5\right].[/latex] Note that the domain and range are always written from lower to higher 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

Figure 9

Find the domain and range of the function [latex]f[/latex] whose graph is shown in Figure 9.

Show Solution

We can observe that the horizontal extent of the graph is –3 but not inclusive to 1 included, so the domain of [latex]f[/latex] is [latex]\left(-3,1\right].[/latex]

The vertical extent of the graph is 0 to –4, so the range is [latex]\left[-4,0\right].[/latex] See Figure 10.

Figure 10

Example 7:  Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function [latex]f[/latex] whose graph is shown in Figure  11.

Figure 11

(credit: modification of work by the U.S. Energy Information Administration)^[2]

Show Solution

The input quantity along the horizontal axis is “years,” which we represent with the variable [latex]t[/latex] for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable [latex]b[/latex] 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 [latex]1973\le t\le 2008[/latex] and the range as approximately [latex]180\le b\le 2010.[/latex]

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.

Figure 12

Show Solution

domain = [1950,2002] range = [47,000,000,89,000,000]

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.

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

For the identity function [latex]f\left(x\right)=x,[/latex] there is no restriction on [latex]x.[/latex] Both the domain and range are the set of all real numbers.

For the absolute value function [latex]f\left(x\right)=|x|,[/latex] there is no restriction on [latex]x.[/latex] However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.

For the quadratic function [latex]f\left(x\right)={x}^{2},[/latex] the domain is all real numbers since any real number can be squared and result in a real number.  We can also see graphically that the horizontal extent of the graph is the whole real number line. We can also see vertically, the graph does not include any negative values for the range, so the range is only nonnegative real numbers.

For the cubic function [latex]f\left(x\right)={x}^{3},[/latex] the domain is all real numbers because any real number raised to the third power will result in a real number.  Graphically we can see that 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.

For the reciprocal function [latex]f\left(x\right)=\frac{1}{x},[/latex] 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. The graph above indicates all other real numbers can be included in the range. In set-builder notation, we could also write [latex]\left\{x|\text{ }x\ne 0\right\},[/latex] the set of all real numbers that are not zero.

For the reciprocal squared function [latex]f\left(x\right)=\frac{1}{{x}^{2}},[/latex] we cannot divide by [latex]0,[/latex] so we must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] 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.

For the square root function [latex]f\left(x\right)=\sqrt[\leftroot{1}\uproot{2} ]{x},[/latex] 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 [latex]x[/latex] is defined to be positive, even though the square of the negative number [latex]-\sqrt[\leftroot{1}\uproot{2} ]{x}[/latex] also gives us [latex]x.[/latex]

For the cube root function [latex]f\left(x\right)=\sqrt[\leftroot{1}\uproot{2}3]{x},[/latex] the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). We can see from the graph that we have not restrictions either horizontally or vertically.

Example 10:  Finding the Domain and Range

Find the domain and range of [latex]f\left(x\right)=2\sqrt[\leftroot{1}\uproot{2} ]{x+4}.[/latex]

Show Solution

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

[latex]x+4\ge 0[/latex] when [latex]x\ge -4[/latex]

The domain of [latex]f\left(x\right)[/latex] is [latex]\left[-4,\infty \right).[/latex]

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

Analysis

Figure 13 represents the function [latex]f.[/latex]

Figure 13

Graphing Piecewise-Defined Functions (Optional)

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 [latex]f\left(x\right)=|x|.[/latex] 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.

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

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 [latex]S[/latex] would be [latex]0.1S[/latex] if [latex]S\le 10\text{,}000[/latex] and [latex]1000+0.2\left(S-10\text{,}000\right)[/latex] if [latex]S>10\text{,}000.[/latex]

Example 12:  Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, [latex]n,[/latex] to the cost, [latex]C.[/latex]

Show Solution

Two different formulas will be needed. For n-values under 10, [latex]C=5n.[/latex] For values of [latex]n[/latex] that are 10 or greater, [latex]C=50.[/latex]

[latex]C\left(n\right)=\bigg\{\begin{array}{ccc}5n& \text{if}& 0<n<10\\ 50& \text{if}& n\ge 10\end{array}[/latex]

Analysis

The function is represented in Figure 14. The graph is a diagonal line from [latex]n=0[/latex] to [latex]n=10[/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10,[/latex] but not all piecewise functions have this property.

Figure 14

Example 13:  Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, [latex]C,[/latex] in dollars for [latex]g[/latex] gigabytes of data transfer.

[latex]C\left(g\right)=\bigg\{\begin{array}{ccc}25& \text{if}& 0<g<2\\ 25+10\left(g-2\right)& \text{if}& g\ge 2\end{array}[/latex]

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

Show Solution

To find the cost of using 1.5 gigabytes of data, [latex]C\left(1.5\right),[/latex] we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

[latex]C\left(1.5\right)=25[/latex][latex]\\[/latex]

To find the cost of using 4 gigabytes of data, [latex]C\left(4\right),[/latex] we see that our input of 4 is greater than 2, so we use the second formula.

[latex]C\left(4\right)=25+10\left(4-2\right)=45[/latex]

Analysis

The function is represented in Figure 15. We can see where the function changes from a constant to a linear  function with slope 10 at [latex]g=2.[/latex] We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

Figure 15

Example 14:  Graphing a Piecewise Function

Sketch a graph of the function.

[latex]f\left(x\right)=\bigg\{\begin{array}{ccc}{x}^{2}& \text{if}& x\le 1\\ 3& \text{if}& 1<x\le 2\\ x& \text{if}& x>2\end{array}[/latex]

Show 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 16 shows the three components of the piecewise function graphed on separate coordinate systems.

Figure 16 (a) [latex]f\left(x\right)={x}^{2}\text{ if }x\le 1; [/latex] (b) [latex]f\left(x\right)=3\text{ if 1< }x\le 2; [/latex] (c) [latex]f\left(x\right)=x\text{ if }x>2[/latex]

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 17.

Figure 17

 

Analysis

Note that the graph does pass the vertical line test even at [latex]x=1[/latex] and [latex]x=2[/latex] because the points [latex]\left(1,3\right)[/latex] and [latex]\left(2,2\right)[/latex] are not part of the graph of the function, though [latex]\left(1,1\right)[/latex] and [latex]\left(2,\text{ }3\right)[/latex] are.

Try it #8

Graph the following piecewise function.

[latex]f\left(x\right)=\bigg\{\begin{array}{ccc}{x}^{3}& \text{if}& x<-1\\ -2& \text{if}& -1<x<4\\ \sqrt{x}& \text{if}& x>4\end{array}[/latex]

Show Solution