Inverse Functions

• Let

What happens when we compose f and g ?

Notice that one function ‘undoes’ the other, producing a final answer of x in both cases.

We say that f and g are inverses of each other.

• Definition : Two functions f and g are inverses of each other if

and

• example : Verify that and are inverses of each other.

• Notation : The inverse of function f is denoted by
• In the example above, we could write and

• We can also restate the definition of inverse functions as

and

• example : Verify that if , then

• Result : The graphs of f and are reflections of each other about the line y = x.

• example : Graph the functions f and in the example above.

• How do we find the inverse of a function?

• The algorithm:

1. Write the equation y = f ( x ) that defines the function.

2. Interchange the variables x and y .

3. Solve for

• example : Find the inverse of y = 3x + 2

1. y = 3x + 2

2. x = 3y + 2 ( interchange x and y )

3. ( solve for )

So,

• example : Find the inverse of f (x) = 4 − 2x

so,

• Does every function have an inverse function?

• To answer this question, think about the fact that the graphs of f and   are reflections of each
other about the line y = x.

• Horizontal Line Test : A function has an inverse if no horizontal straight line intersects its graph
more than once. The function is said to be one-to-one.

• example : Does y = x² have an inverse?

• example : Does y = x² , x ≥ 0 have an inverse?