Composition of functions (§6.1)

A function takes one number and gives you another. 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.

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).

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!

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