Inverse Functions

A function is one-to-one (1-1) if it does not assign the same value to two different elements of its domain:

If , then .

If f is a 1-1 function, then it has an inverse function defined by iff , for all in the range of f.

The domain of is the range of , and the range of is the domain of .

To find a formula for , we can

1. Set .

2. Solve for in terms of , if possible.

3. Set .

[Another common way to do this is to

1. Set , and then interchange and .

2. Solve for in terms of , if possible.

3. Set .]

Ex 1 Show whether or not the function is one-to-one.

Sol , so is a 1-1 function.

Ex 2 Show whether or not the function is one-to-one.

Sol Setting , for example, and solving gives that ; so is not a 1-1 function.

Ex 3 If , find a formula for

Sol Let . Then , so or . Thus , so .

Pr 1 If , find a formula for .

Pr 2 If , find a formula for .

Pr 3 If , find a formula for and find the domain for .

Pr 4 If , find a formula for .

Pr 5 Show whether or not the function has an inverse.

Pr 6 Let for . Find a formula for and find the domain for .

Pr 7 If for , find .

Pr 8 If , find a formula for and find the domain for .