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
The −1 in
f−1 is not an exponent!
Do not misinterpret the −1 in
Given the function f described by
To show that f is one-to-one, we show that if
Thus, if
Now Try Exercise 17.
Given the function g described by
We can prove that g is not one-to-one by finding two numbers a and b for which
Now Try Exercise 21.
The following graphs show a function, in blue, and its inverse, in red. To determine whether the inverse is a function, we can apply the vertical-line test to its graph. By reflecting each such vertical line across the line
From the graphs shown, determine whether each function is one-to-one and thus has an inverse that is a function.
For each function, we apply the horizontal-line test.
RESULT | REASON |
---|---|
|
No horizontal line intersects the graph more than once. There are many horizontal lines that intersect the graph more than once. Note that where the line No horizontal line intersects the graph more than once. There are many horizontal lines that intersect the graph more than once. |
Now Try Exercises 25 and 27.