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Inverse Functions

One-to- One Functions and Their Inverses

Let f be a function with domain A. f is said to be one-to-one if no two
elements in A have the same image.

Example 1: De termine if the fol lowing function is one-to-one.

•A one-to-one function has an inverse function.
•The inverse function reverses whatever the first function did.

Example: The formula

is used to convert from x degrees Celsius to y degrees Fahrenheit. The formula


is used to convert from x degrees Fahrenheit to y degrees Celsius

•The inverse of a function f is denoted by f-1 , read “f-inverse”.

Example 2: As sume that the domain of f is all real numbers and that f is one -to-one. If

f(7) = 9 and f(9) = -12, then what is

Example 3:
If f and g are inverse functions, f (-2) = 3 and f (3) = -2 . Find g(-2) .

Domain and Range:

The domain of f is the range of f-1 and the range of f is the domain of f-1 .

These two statements mean exactly the same thing:

1. f is one-to-one (1-1)
2. f has an inverse function

Property of Inverse Functions

Let f and g be two functions such that ( f o g)(x) = x for every x in the domain of
g and (g o f )(x) = x for every x in the domain of f then f and g are inverses of
each other.

Example 4:
Show that the following functions are inverses of each other

How to find the inverse of a function: (if it exists!)

1. Replace “ f (x) ” by “y”.
2. Ex change x and y .
3. Solve for y .
4. Replace “y” by “ f-1(x) ”.
5. Verify!

Example 5:
Find the inverse function of f (x) = 2x - 7 .

Example 6:
Assume f (x) is a one-to-one function. Find the inverse function f-1(x) given that

Example 7:
Find the inverse function f-1(x) given that

Example 8:
Assume g(x) is a one-to-one function. Find the inverse function g-1(x) given that

Example 9: Assume g(x) is a one-to-one function. Find the inverse function g-1(x) given that

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