The Lesson

A composite function is a function of a function. It combines two or more functions so that the output of one function becomes the input of another. Evaluating a composite function means putting an input into a composite function, and finding the output it relates to.

Understanding Evaluating a Composite Function

To evaluate a composite function means to see what output an input is mapped to. The image below shows a mapping diagram of a composite function which relates a set of inputs to a set of outputs. If we wanted to evaluate the function when the input is 2, we would see which output it is mapped to.

How to Evaluate a Composite Function

Evaluating a composite function is easy.

Question

Two functions are f(x) = 2x + 1 and g(x) = x + 2. Evaluate the composite function fg(x) at x = 2.

There are two methods for evaluating the composite function.

Method 1

This method is simpler.

Step-by-Step:

1

Understand the composite function. In our example, the composite function is fg(x). Reading right to left, this means:
  • the input x is passed into the function g
  • which is passed into the function f
Because we are evaluating the function at x = 2, we will pass in 2 as an input, rather than x.

2

Pass the input 2 into the function g. This is evaluating the function g(x) at x = 2. Substitute x = 2 into g(x).
g(x) = x + 2 g(2) = 2 + 2 g(2) = 4

3

Pass the g(2) into the function f. We have found in Step 2 that g(2) = 4. Evaluate the function f(x) at x = 4 by substituting x = 4 into f(x).
f(x) = 2x + 1 f(4) = 2 × 4 + 1 f(4) = 8 + 1 f(4) = 9

Answer:

The composite function fg(x) evaluated at x = 2 is: fg(2) = 9.

Method 2

This method is more complicated because you find the composite function before evaluating it. The advantage is that you can then evaluate the composite function at many different values.

Step-by-Step:

1

Find the composite function fg(x).
fg(x)Find the left most letter. It is f
f(x) = 2x + 1Write out the function f(x)
fg(x) = 2g(x) + 1Insert a g to the right of the f in the function name and replace x with g(x)
fg(x) = 2(x + 2) + 1Substitute g(x) = x + 2 into the function (put it in brackets)
fg(x) = 2x + 4 + 1Expand the brackets
fg(x) = 2x + 4 + 1Collect the constant terms
fg(x) = 2x + 5
The composite function fg(x) = 2x + 5.

2

Evaluate the composite function fg(x) at x = 2 by substituting x = 2 into fg(x).
fg(x) = 2x + 5 fg(2) = 2 × 2 + 5 fg(2) = 4 + 5 fg(2) = 9

Answer:

The composite function fg(x) evaluated at x = 2 is: fg(2) = 9. Both methods give the same answer: if the functions f(x) = 2x + 1 and g(x) = x + 2 are combined into a composite function fg(x) and evaluated at 2, the answer is 9.

Lesson Slides

The slider below shows another real example of how to evaluate a composite function. Open the slider in a new tab

A Note on Notation

If a composite function fg is evaluated at a number (e.g. 2), this is denoted fg(2).