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 .

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