Section 2.5 Inverse Functions
f-(x) = 2x+4
The function f(x) has an inverse f--1(x).
f--1(x) = ½ x-2
One way of thinking about inverse functions.
The inverse function switches x and y.
Example:
f(x) = y = 2x+4
The inverse of f(x) is:
x = 2y+3
x-3 = 2y
y = (x-4)/2 = ½ x – 2
Another way of thinking about inverse functions.
The inverse function of f(x) undoes what f(x) does.
f(x)
x
-2
-1
0
1
2
y
0
2
4
6
8
Mapping from x  y
-2  0
-1  2
04
16
28
(an input  an output)
The inverse would
take each output and
send it back to the
input.
0  -2
2  -1
40
61
82
Horizontal rule
Recall that every function passes the vertical line test.
For every x, there is only one y.
(For every input, there is only one output.)
A function has an inverse only if it passes the horizontal line test.
For every y, there is only one x.
(For every output, there is only one input giving that output.)
f--1 (x)
x
0
2
4
6
8
y
-2
-1
0
1
2
One-to-one
If a function passes both the vertical and horizontal line tests, this means that each input
gives exactly one output and each output has exactly one input. We call this function
“one-to-one”.
A function that is one-to-one has an
inverse.
A function but not one-to-one: