Composition of Functions

Composition of Functions Composition is when we take the output of one function and use it as the input for another function. Another way to state this is h(x) = f(g(x)), or:




Note: This is pronounced "f of g of x", not "fog x".

Let's take a look at an example of composition. For this example, we will use:

Composition is defined as:

First, we fill in f and g:

Then, we simplify, getting:

To find the domain of a composite function, we need to first find the domain of g(x). Then, since we are putting the results of g(x) into f(x), we need to find which results of g(x) are in the domain of f(x). If x is in the domain of g(x), and g(x) is in the domain of f(x), then x is in the domain of h(x).

For example:

Filling in f and g, we get:

Since the denominator of g(x) cannot equal 0, x cannot equal 5. Therefore, the domain of g is (-∞, 5) ∪ (5, ∞). But we're not done yet. We cannot have a negative under the square root in f, so we can exclude numbers less than -3 from the domain of f. But, since this is a composite function, g(x) is the domain of f, so g(x) cannot be less than -3. By excluding all x that would make g(x) less than -3, we have the domain of (f ○ g) (x), which is:

Things to Remember:
1. (f ○ g) (x) is pronounced "f of g of x", not "fog x".
2. Always remember to put parentheses in the right places.