# What is a Function? | Math

February 11, 2015
Functions relate an input to an output. In mathematical equations, a function is typically written as “f(x) = …” There are other ways to express functions, but you should keep this expression in mind; functions are a high emphasis area on the TASC Test Assessing Secondary Completion™ Math subtest. There are many things you can do with a function and many different ways to express functions. Here, we’ll focus on the basics. First, you’ll need to know about relations. According to Elizabeth Stapel of Purple Math, a relation is “a relationship between sets of information. Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. In relations and functions, the pairs of names and heights are “ordered,” which means one comes first and the other comes second.” The set of all the starting points (the numbers that come first) is called the domain, and the set of all the ending points (the second set of numbers) is the range. The domain is what you start with, and the range is what you end with – or, to put it another way, the domain is the x and the range is the y. Stapel calls functions “well-behaved relations” because when you are given a starting point, you know exactly where to go. Functions are single-valued. Whenever you’re given the x, you can find the y. You may see functions expressed in a chart, such as:
 X: x Y: x2 0.5 0.25 2 4 5 25
You may also see them expressed in ordered pairs. In these pairs, the input is listed first and the output is listed second: (input, output). Or, put differently: (x, f(x)). A function can then be defined as a set of ordered pairs. For example, a function can be expressed in this way: {(3,5), (6,8), (11,4)} This example says 3 is related to 5, 6 is related to 8, and 11 is related to 4. The domain is {3,6,11}. The range is {5,8,4}. Functions are similar to graphs and inequalities, so keep those tips and tricks in mind while you’re working through these functions, and keep practicing.
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