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3.2+Domain and Range 205 3.2 Domain and Range Learning Objectives In this section, you will: > Find the domain of a function defined by an equation. > Graph piecewise-defined functions. Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s—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. ‘Top-Five Grossing Horror Movies for years 2000-2013 Market Share of Horror Moveis, by Year g & 2 Inflation-adjusted Gross, in Millions of Dollars o8 BS4S88 esxuevruar Sx 228 22 3 fob ob bot L Tam Hannibal The The The Sy Sy ob oh 0 oo fh Ds WY AD Legend (2001) Ring Grudge Conjuring SPP Sy SM Ge My ps (2007) (2002) (2004) (2013) Figure 1 Based on data compiled by www.the-numbers.com.* Finding the Domain of a Function Defined by an Equation ind Function N ‘ion, 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. Domain Function’ Range machine SEN -LY 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, they would need to express the interval that is more than 0 and less than or equal to 100 and write (0, 100].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 3. 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 
