5.1 Inverse Functions

  • Determine whether a function is one-to-one, and if it is, find a formula for its inverse.

  • Simplify expressions of the type (f∘f−1)(x) and (f−1∘f)(x).

Inverses

When we go from an output of a function back to its input or inputs, we get an inverse relation. When that relation is a function, we have an inverse function.

Consider the relation h given as follows:

h={(−8, 5), (4, −2), (−7, 1), (3.8, 6.2)}.

Suppose we interchange the first and second coordinates. The relation we obtain is called the inverse of the relation h and is given as follows:

Inverse of h={(5, −8), (−2, 4), (1, −7), (6.2, 3.8)}.

Example 1

Consider the relation g given by

g={(2, 4), (−1, 3), (−2, 0)}.

Graph the relation in blue. Find the inverse and graph it in red.

Solution

The relation g is shown in blue in the figure at left. The inverse of the relation is

{(4, 2), (3, −1), (0, −2)}

and is shown in red. The pairs in the inverse are reflections of the pairs in g across the line y=x.

Now Try Exercise 1.

Example 2

Find an equation for the inverse of the relation

y=x2−5x.

Solution

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

x=y2−5y.

Now Try Exercise 9.

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.

x=y2−5y y
6 −1
0 0
−6 2
−4 4
x y=x2−5x
−1 6
0 0
2 −6
4 −4
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