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 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.

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 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.

Example: Finding the Domain of a Function Involving a Denominator (Rational Function)

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

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{cases}2-x=0\hfill \\ -x=-2\hfill \\ x=2\hfill \end{cases}[/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].

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

Watch the following video to see more examples of how to find the domain of a rational function (one with a fraction).

The next video gives more examples of how to define the domain of a function that contains an even root.

Try it now

When you are defining the domain of a function, it can help to graph it, especially when you have a rational or a function with an even root.

First, determine the domain restrictions for the following functions, then graph each one to check whether your domain agrees with the graph.

1. [latex]f(x) = \sqrt{2x-4}+5[/latex]
2. [latex]g(x) = \frac{2x+4}{x-1}[/latex]

Next, use Desmos to evaluate your function at the domain restriction you found. What function value does Desmos give you?