Inverse Functions
As we have seen, a function is a rule that assigns exactly one output to
each input. A function and its inverse function are related in the following
ways:
• the input of the function is the same as the output of its inverse function;
• the output of the function is the same as the input of its inverse function.
For example, let’s examine the functions
Here is a table of ordered pairs that satisfy each function and their graphs:
Note:
Notice that the function f(x) = 3x multiplies each input by 3 while the
function
does the reverse; that
is, it divides each input by 3.
Be careful! The -1 in f -1(x) is a notation
used to indicate an inverse function. It is
not an exponent.
That is,
Example 1
The graph of a function f(x) is shown. Fill in the table for f -1(x).
Solution
The inputs for f -1 are 3, 4, and 6. These, then, must be the
outputs for f.
Thus, we first identify order pairs for f with outputs of 3, 4, and 6.
For example, from the graph we can see that when the y-value of f is 3 the
corresponding x-value is -2.
In this way, we can use the graph to find all three ordered pairs for f. They
are (-2, 3), (0, 4), and (4, 6).
To complete the table for f
-1, recall that the y-values for f
-1 are the same
as the x-values for f.
