Analysis

Finding the Domain and Range of Functions

You just got a new part-time job at the mall that pays a base rate of $150/week plus $5/sale. Your boss encourages you to make as many sales as possible but she will cap your weekly earnings at $250. What are the domain and range of the function represented by this situation?

Domain and Range of a Function

The input and output of a function is also called the domain and range. The domain of a function is the set of all input values. The range of a function is the set of all output values. Sometimes, a function is a set of points. In this case, the domain is all the x− values and the range is all the y− values. Functions can also be linear and polynomial equations. In these instances, you need to graph the function to see where it is defined. You may notice that some functions are defined for “all real numbers.” The symbol, ℝ, is used to denote the set of all real numbers.

Mathematically, this would be written: x∈(−∞,−2]∪(−1,∞). The ∪ symbol means “union.” In words, the domain is “all real numbers except those between -2 and -1.” Notice that -2 is included in the domain because the dot at -2 is closed. To find the range, we need to look at the possible y−values. Changing our viewpoint to look at the y−axis, at first glance, it looks like the function is not defined from 1 to -3.

However, upon further investigation, the branch on the left does pass through the yellow region, where we though the function was not defined. This means that the function is defined between 1 and -3 and thus for all real numbers. However, below -3, there are no y−values. The range is y∈[−3,∞).

Earlier, you were asked to find the domain and range of the function of your sales, where you make a base rate of $150/week plus $5/sale (your weekly earnings are capped at $250/week).

Solution

The function represented by this situation can be written as y=150+5x, where x is the number of sales you make. You can’t make a negative number of sales, so the least amount of sales you can make is zero. To find the maximum number of sales before you reach the cap, we must plug in $250 for y.

250=150+5x