## 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