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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.
If the inverse of the function, f(x) is also a function. It is called Inverse
function of f(x) and is written 1( )f x .
Example:
Graph the function ( ) 2 1f x x and draw the graph of the inverse 1( )f x .
The domain of ( )f x must be equal to the range of
1( )f x and vice versa.
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The inverse of a function algebraically.
Example: Find the inverse of ( ) 2 1f x x
Steps to follow: 1) Replace ( )f x with y y = 2x - 1
2) Interchange x and y x = 2y – 1
3) Solve for y 1
2
xy
4) Replace y with 1( )f x 1 1( )
2
xf x
This can be verified by using the composition of functions:
Let ( ) 2 1f x x and 1
( )2
xg x
( ( ))f g x ( ( ))g f x
12 1
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x
(2 1) 1
2
x
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.
One to one means that each x-value has exactly one unique y-value. And
each y-value corresponds to exactly one x-value.
Applying the horizontal line test will check to see if a function is one to one.
Not a 1-1 so the inverse is not a function
Is a 1-1 so the inverse is a function
