## Finding the Domain of a Function Defined by an Equation

When we purchase something from a retailer, we want to know what kind of payment they will accept. Some online retailers will accept a credit card or Paypal, but not Bitcoin. Similarly, with functions, we cannot always assume that we can evaluate a function using any number as $x$. For example, what if you are working with the function $f(x) = \frac{1}{x}$ and you try to evaluate the function at $x = 0$? You cannot divide by zero, so we must let the user know that $x$ cannot be zero, much like how most retailers do not accept Bitcoin. The set of values that can be used for $x$ in a given function is called its domain, and the resulting values that will be output from the function and called the range. In this section, we will find the domain of a function and express it in many ways.

### Domain and Range

#### Learning Objectives

In this section, you will:

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

If you're 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 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. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

Figure 1 Based on data compiled by www.the-numbers.com.

Source: Rice University, https://openstax.org/books/college-algebra/pages/3-2-domain-and-range