Inverse Functions

In this section, we will learn about inverse functions. Previously we learned that a function is a special type of relation where each x-value is associated with one and only one y-value. A function is allowed to have a y-value that is associated with more than one x-value. For example, the relation:{(1,6),(2,4),(9,6)}is a function that has the same y-value (6), associated with two different x-values: (1, and 9). In order to talk about inverse functions, we must first learn about one-to-one functions. For a function to be one-to-one, each y-value can be associated with one and only one x-value. This means our previous example is a function, but not a one-to-one function. When we have a one-to-one function, we can create a function known as the inverse by interchanging x and y. As an example, the relation: F = {(1,7),(2,0),(-8,-4)}is a one-to-one function. We can find the inverse of F by interchanging the x and y: G = {(7,1),(0,2),(-4,-8)}. F and G are inverses of each other. Most examples will not be given as a set of ordered pairs. How can we take a more complex function and determine if it is one-to-one? We can graph our function and use the horizontal line test. If any horizontal line intersects the graph of a function in more than one location, it is not the graph of a one-to-one function. The reason is simple; no two ordered pairs of a given one-to-one function can have the same y-value. This means, no horizontal line (y = k) will intersect the graph in more than one location. If it does, the same y-value is associated with more than one x-value.