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. 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 ticket 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.

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 [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, consider excluding values that would make the radicand negative.

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

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

See Figure 3 for a summary of interval notation.

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 \le 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 \le 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 or equal to 30.”

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

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 [latex]\left\{x|\text{ }|x|\ge 3\right\}=\left(-\infty ,-3\right]\cup \left[3,\infty \right)[/latex]

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.

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{or}\,x>5[/latex]
Set-builder notation	[latex]\left\{x|1\le x\le 3\,\text{or}\,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 set includes the boundary. Using a parenthesis means the boundary is not included in the set.

Try It

Given Figure 7, specify the graphed set in

1. words
2. set-builder notation
3. interval notation

Show Solution

1. values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
2. {[latex]x | x\le -2\, \  \text{or} \,-1\le x[/latex] <3 };
3. [latex]\left(-\infty ,-2\right]\cup \left[-1,3\right)[/latex]

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 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 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.

Finding Domain and Range from a Graph

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 to 1, 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.

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.

Finding the Domain and Range

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

Show Solution

<p style=”text-align: center”>We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.
<p style=”text-align: center”>[latex]x+4\ge 0\text{ when }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 22 represents the function [latex]f.[/latex]

Access these online resources for additional instruction and practice with domain and range.

Graphing Piecewise-Defined Functions

Sometimes, we come across a function requiring more than one formula 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 in 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 \text{\$}10\text{,}000[/latex] and[latex]\text{\$}1000+0.2\left(S-\text{\$}10\text{,}000\right)[/latex] if [latex]S>\text{\$}10\text{,}000.[/latex]

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]

<p style=”text-align: center”>[latex]C(n)=\bigg\{\begin{array}{l}5n \ \ \ \ \ \text{ if } \ 0 \lt n \lt 10\\ 50 \ \ \ \ \ \ \text{ if } \ \ n \ge 10\end{array}[/latex]

Analysis

The function is represented in Figure 23. 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.