A function is one-to-one if different input values always produce different output values. In other words, a function is one-to-one if no -values are reused.

Examples

is not one-to-one because . The value is associated with two different -values, and .

is one-to-one because no two different numbers have the same square root.

A function is one-to-one if and only if its graph passes the horizontal line test. (Remember that the graph of any function must pass the vertical line test.)

Let . Is a one-to-one function?

Is one-to-one? What about ?

Inverse Functions

The inverse of a function is a new function that undoes the action of the original function.

Examples

and are inverses. What one does, the other undoes.

Not all functions have inverse functions. For example, the constant function does not have an inverse.

One-to-one functions have inverses.

Formal definition of Inverse: The functions and are inverses if for all in the domain of and for all in the domain of .

When is the inverse of , we write .

If exists, then

and are one-to-one functions,

the domain of is the range of ,

the range of is the domain of ,

for all in the domain of ,

for all in the domain of .

The last two bullet points above reiterate the fact that inverse functions undo the action of each other: if and only if .

VERY IMPORTANT IDEA: If is a point on the graph of , then is a point on the graph of . Since is the reflection of about the line , it follows that the graph of can be obtained from that of by reflecting it about the line .

Examples

Show that and are inverses.

Let . Compute .

Find the inverse of the function defined by .

Finding Inverses

For some functions, we can use algebra to determine inverse functions. Essentially, to find an inverse function we "undo" the original function.