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.
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.)
The inverse of a function is a new function that undoes the action of the original function.
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
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 .
For some functions, we can use algebra to determine inverse functions. Essentially, to find an inverse function we "undo" the original function.