Inverse functions (§6.2)

As composition involves performing one function after another, so inverses involve performing a function backwards.

Two functions f and g are inverse functions of each other if:

Not every function has an inverse function! If f does have an inverse function, then it only has one; we call f one-to-one and denote its unique inverse function by f−1.

Warning: f−1 does not mean 1/f!

There are two ways to caclulate f−1:

If you ever get more than one solution when you solve the equation, then f is not one-to-one, and it has no inverse.

We have:

The graphs of inverse functions are related by switching x and y. In particular:

A function is one-to-one if and only if its graph satisfies the Horizontal Line Test: every horizontal line goes through the graph at most once.
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