# Composition of functions

If we have two or more functions that are contained within each other, we call it compound functions. Its general expression is as follows:

The value of the compound function in x is equal to the function f evaluated in g (x).

## Definition

Given two functions f and g, the composite function of f with g is defined as:

his domain is the set of all x (variable) that are in the domain of g, such that g (x) belongs to the domain of f. That is to say:

Exercise 1: Sean:

y

Determine:

We know that:

We first substitute the value of g (x):

Now, to evaluate f, we substitute x = g (x):

We simplify:

We solve the binomial:

To determine the domain we must remember that:

We will first look for the domain of g (x); Since the function has a root and the variable is inside it, it cannot be negative:

Then the domain of g will be all values greater than or equal to four, that is:

For the domain of f: since its function is a polynomial, it will be all real numbers, that is:

Since we are looking for the domain of the compound function, we know that this domain depends directly on g (x), therefore: