The inverse of a function is obtained by reflecting the graph of the original function over the line y=x. The inverse of a function f(x) is written as f^-1(x). For a function to have an inverse, it must be one-to-one, meaning each x-value only corresponds to a single y-value. To check if a function is one-to-one, apply the horizontal line test - if no horizontal line intersects the graph at more than one point, it is one-to-one and will have an inverse function.

1. Lesson 52
Inverse Functions
Text: Chapter 2, section 6 & 7
 The inverse of a relation is the set of ordered pairs obtained by
interchanging the coordinates of each ordered pair. ( x, y) ( y, x)
 The graph of an inverse function is the reflection of the graph over the line
y=x.
2– 2
10 10
8 8
6 6
4 4
2– 2
10 10
8 8
6 6
4 4  If the inverse of the function, f(x) is also a function. It is called Inverse
function of f(x) and is written f 1 ( x) .
 Example:
Graph the function f ( x) 2 x 1 and draw the graph of the inverse f 1 ( x) .
y
10
8
6
4
2 2– 2
10 10
8 8
6 6
4 4
2– 2
10 10
8 8
6 6
4 4
– 10 – 8 – 6 – 4 – 2 2 4 6 8 10 x
– 2
– 4
– 6 y
10
– 8 8
– 10 6
4
2
The domain of f ( x) must be equal to the range of
– 10 – 8 – 6 – 4 – 2 2 4 6 8 10 x
f 1 ( x) and vice versa. – 2
– 4
– 6
– 8
– 10

2.  The inverse of a function algebraically.
Example:
Find the inverse of f ( x) 2x 1
Steps to follow:
1) Replace f ( x) with y y = 2x - 1
2) Interchange x and y x = 2y – 1
x 1
3) Solve for y y
2
x 1
4) Replace y with f 1 ( x) f 1 ( x)
2
 This can be verified by using the composition of functions:
x 1
Let f ( x) 2 x 1 and g ( x)
2
f ( g ( x)) g ( f ( x))
x 1 (2 x 1) 1
2 1
2 2
x x
Since both compositions produce x, f(x) and g(x) are inverses.
 Every inverse is not a function. If the original function is one-to-one
function then its inverse is also a function.
2– 2
10 10
8 8
6 6
4 4
 One to one means that each x-value has exactly one unique y-value. And
2– 2
10 10
8 8
6 6
4 4
2– 2 2– 2
10 10
8 8
6 6
4 4
each y-value corresponds to exactly one x-value.
10 10
8 8
6 6
4 4
 Applying the horizontal line test will check to see if a function is one to
one. y y
10
Not a 1-1 so 10
Is a 1-1 so
8 8
the inverse is the inverse
6 6
4
not a function 4
is a function
2 2
– 10 – 8 – 6 – 4 – 2 2 4 6 8 10 x – 10 – 8 – 6 – 4 – 2 2 4 6 8 10 x
– 2 – 2
– 4 – 4
– 6 – 6
– 8 – 8
– 10 – 10