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:

f(g(x)) = x whenever x ∈ dom g, and
g(f(x)) = x whenever x ∈ dom f.

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:

Start with x = f(y) and solve for y; this gives you y = f⁻¹(x).
Start with y = f(x) and solve for x; this gives you x = f⁻¹(y).

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:

f⁻¹(f(x)) = x whenever x ∈ dom f, and
f(f⁻¹(y)) = y whenever y ∈ ran f.

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

The range of f is the same as the domain of f⁻¹;
the range of f⁻¹ is the same as the domain of f.

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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