Section 2.6 – Inverse Functions
DOES an inverse function exist?
IF YES, you can find the inverse function.
The Existence of the Inverse of f(x)
IF for every x there is at most one y (function)
AND
IF for every y there is at most one x (one-to-one)
then an inverse function of f(x) exists.
The inverse function is denoted by f 1  x 
Graphical Existence of Inverse
Passes BOTH vertical and horizontal line test.
Inverse Exists
No Inverse Exists
(1, 1), (-1, 1)
No Inverse Exists
(1, 0), (0, 0)
Inverse Exists
No Inverse Exists
(3, 0.9), (7, 0.9)
Inverse Exists
Inverse Exists
No Inverse Exists
(-3, 2), (0, 2)
No Inverse Exists
(2, -2), (2, 5)
Does an inverse exist?
January
February
March
July
Winter
Spring
Summer
No Inverse Exists
(Jan, Winter), (Feb, Winter)
Ford
President
Bush
Vice-President
Carter
Clinton
No Inverse Exists
(Ford, President),
(Ford, Vice-President)
t
f(t)
0
5
1
2
2
2
3
4
No Inverse Exists
(1, 2), (2, 2)
 2, 4, 3, 4 , 1, 5 
No Inverse Exists
(2, 4), (3, 4)
 6, 5, 8,  1,  6, 3 
No Inverse Exists
(6, 5), (6, 3)
Algebraic Existence of Inverse
y  x2  6x  4
No Inverse Exists
(-4, -12), (-2, -12)
xy 7
x  y2
No Inverse Exists
(4, 2), (4, -2)
Inverse Exists
x  y2  4
x2  y2  16
No Inverse Exists
(0, 2), (0, -2)
No Inverse Exists
(4, 0), (-4, 0)
FINDING the inverse which exists
1. Switch the x and the y.
2. (Algebraically) Solve for y.
3. Replace y with f 1  x 
Finding the inverse function GRAPHICALLY
Finding the Inverse Function TABULARLY
x -2
f(x) 0
3
-1
5
4
7
9
8
-3
9
2
x
-1
3
4
5
9
7
-3
8
2
9
0
f 1  x  -2
x
f(x)
2
-7
3
-1
4
3
5
4
6
8
7
4
x
f(x)
2
-7
3
-1
4
3
5
4
6
8
7
4
No Inverse Exists
(5, 4), (7, 4)
Finding the Inverse ANALYTICALLY
f  x   2x  3
x  2y  3
x3
y
2
x3
 f 1  x 
2
xy 7
yx7
y 7x
f 1  x   7  x
x  y 1
x2  y  1
x2  1  y
x 2  1  f 1  x 
f  x   x3  3
f x  3 x  4
x  y3  3
x  3 y 4
x4 3 y
x  3  y3
3
f x  x  1
x3  f
1
x
 x  4  y
3
 x  4   f 1  x 
3