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.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2016 April 25. Toby reserves no legal rights to it.

The permanent URI of this web page is http://tobybartels.name/MATH-1150/2020SP/inverses/.