Write equations of two functions f and g such that f ° g = g ° f = x. (In Section we will...

Write equations of two functions f and g such that f ° g = g ° f = x. (In Section we will study inverse functions. If f ° g = g ° f = x, functions f and g are inverses of each other.)

Section

Find an equation for the inverse of the relation

y = x2 − 5x.

We interchange x and y and obtain an equation of the inverse:

x = y2 − 5y

If a relation is given by an equation, then the solutions of the inverse can be found from those of the original equation by interchanging the first and second coordinates of each ordered pair. Thus the graphs of a relation and its inverse are always reflections of each other across the line y = x. This is illustrated with the equations of Example 2 in the tables and graph below. We will explore inverses and their graphs later in this section.

Inverses and One-to-One Functions

Let’s consider the following two functions.

Suppose we reverse the arrows. Are these inverse relations functions?

We see that the inverse of the postage function is not a function. Like all functions, each input in the postage function has exactly one output. However, the output for 2009, 2010, and 2011 is 44. Thus in the inverse of the postage function, the input 44 has three outputs, 2009, 2010, and 2011. When two or more inputs of a function have the same output, the inverse relation cannot be a function. In the cubing function, each output corresponds to exactly one input, so its inverse is also a function. The cubing function is an example of a one-to-one function.

If the inverse of a function f is also a function, it is named f− 1 (read “f-inverse”).

The − 1 in f − 1 an exponent!

Do not misinterpret the − 1 in f− 1 as a negative exponent: f− 1 does not mean the reciprocal of f and f− 1(x) is not equal to