Function Composition Examples

By Robert O

What is the definition of composite functions?

The basic definition of a composite function is that it has two functions in one. The inner function is a variable to the outer function.


is an example of a composite function. The inner function is


while the outer function is


which is equal to f(x).

What is the notation of composite functions?

The general notation for composite function is (f o g)(x) or f(g(x)), read as f of g of x. Using this notation, we say that g(x) is a function of f(x).

When solving composite functions, you have to be very careful not to reverse that order. In many cases,

    \[f(g(x)) \neq g(f(x))\]

Don’t get it wrong when evaluating composite functions. Once you know the order, then evaluating a composite function becomes as easy as evaluating any other functions.

Steps for evaluating composite functions

We use three steps when evaluating composite functions.

  • Step 1: Rewrite the function into a different form that is easy to understand. For example, (f o g)(x) becomes f(g(x)).
  • Step 2: Substitute for x using the value given beginning with the inner function then evaluating the outer function.
  • Step 3: Simplify the result if possible.

To better understand the steps, we will illustrate using a few examples in the following section.

Example 1:

Given f(x) = – 4 x + 9 and g(x) = 2 x – 7 what is f(g(x)).


In the example above, we have the components of a composite function. The next step is to rewrite them as a composite function.

    \[f(g(x))=-4(2 x-7)+9\]

just substituting function g for x in function f.

By expansion and simplification:

    \[f(g(x))= -8 x+37\]

Example 2:


    \[f(x)=3 x-5\]


    \[g(x)=2 x^{2}-7 x\]

what is f(g(x)).


Again, we have to components of a composite function. We use the same steps to evaluate it.

    \[f(g(x))=3\left(2 x^{2}-7 x\right)-5\]

just substituting function g for x in function f.

    \[f(g(x))=6 x^{2}-21 x-5\]

Example 3:

Given that f(x) = – 6 x + 5 and h(x) = – 9 x – 11 find (f o h)(2).


This is a different problem from the previous two examples. We have function components that we have to combine and then evaluate the composite function at x = 2.

    \[f(h(x))=-6(-9 x-11)+5\]

just substituting function h for x in function f.

Expanding and simplifying:

    \[\mathrm{f}(\mathrm{h}(\mathrm{x}))=54 \mathrm{x}+71\]

Next, we substitute for x=2:

    \[\mathrm{f}(\mathrm{h}(\mathrm{x}))=54 \times 2+71\]


Example 4:

Evaluate (g o h)(6) given g(x) = 5 x + 4 and h(x) = x – 3


Just like the previous example, we need to evaluate the value of the function at x = 6.

We start by forming a composite function from its components:


just substituting function h for x in function g.

    \[g(h(x))=5 x-15+4\]

    \[\mathrm{g}(\mathrm{h}(\mathrm{x}))=5 \mathrm{x}-11\]

Next, we evaluate the function at x=6:

    \[\mathrm{g}(\mathrm{h}(6))=5 \times 6-11\]



When forming a composite function from its components, the result may be a quadratic expression or any other polynomial. Just remember that the procedure does not change. Follow the steps to evaluate the function the way you do with other functions.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.