Evaluating a Composite Function
(KS4, Year 10)

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.


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.



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.


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


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


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.



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.


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


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).
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See Also

What is a function? Expanding brackets Collecting like terms