Composite and inverse functions (§3.4, §3.7)

A function takes one number as input and gives you another as output. You can then take this output and use it as the input to another function, to get yet another number. This is composition of functions.

Definition

If f and g are functions, then their composite is also a function, denoted f ∘ g, which I usually read as ‘f after g’. It may be defined as follows:

(f ∘ g)(x) = f(g(x)).

Notice that you do g first and f afterwards. The two composites f ∘ g and g ∘ f are not usually the same.

Sometimes it helps to use different variables for the different functions. That is:

Let y be g(x);
let z be f(y);
then z = f(g(x)) = (f ∘ g)(x).

Domains of composites

If f and g are both defined for every real number, then so is f ∘ g. But in general, the domain of f ∘ g is part of the domain of g. We have:

dom (f ∘ g) = {x | x ∈ dom g, g(x) ∈ dom f}; that is,
x belongs to the domain of f ∘ g if and only if x belongs to the domain of g and g(x) belongs to the domain of f.

If you simplify the expression for f ∘ g, then this will only help with the second condition; be sure to go back and check the domain of g!

Inverse functions

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!; in fact, it is precisely the one-to-one functions that have them. If f is one-to-one, then it only has one inverse function; we write this unique inverse function as 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.

Graphs of inverse functions

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.

This is why 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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