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Finding a Composite Function
(KS4, Year 10)

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A composite function is a function of a function. A composite function combines two or more functions so that the output of one function becomes the input of another.

How to Find a Composite Function

Finding a composite function is easy.

Question

Find the composite function fg(x), where f(x) = x2 + 1 and g(x) = x + 2.

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
find composite function step 1

2

Find the left most letter. In our example, the composite function is fg(x). f is the left most letter. Write out the function f(x)
f(x) = x2 + 1

3

We are passing g(x) into this function
  • Insert a g to the right of the f in the function name.
    fg(x) = x2 + 1
  • Replace x with g(x).
    fg(x) = g(x)2 + 1

4

Substitute g(x) = x + 2 into the function.
fg(x) = (x + 2)2 + 1
Don't forget: Use brackets when substituting an expression. This reminds you that the whole expression is being squared.

5

Simplify the equation.
fg(x) = (x + 2)2 + 1
fg(x) = (x + 2)(x + 2) + 1 (x + 2)2 = (x + 2) × (x + 2)
fg(x) = x2 + 4x + 4 + 1 Use the FOIL method to expand the brackets
fg(x) = x2 + 4x + 4 + 1 Collect the constant terms
fg(x) = x2 + 4x + 5

Answer:

The composite function fg(x) of f(x) = x2 + 1 and g(x) = x + 2 is: fg(x) = x2 + 4x + 5.

Lesson Slides

The slider below shows another real example of how to find a composite function.

A Note on Notation

The image below shows a composite function gf(x), where a function f is passed to a function g: composite_function_mini The composite function is gf(x).
  • f is applied to the input x.
  • g is applied to the function f.
The composite function can also be denoted (g ∘ f)(x) or g(f(x)). If the functions are applied in the opposite order... composite_function_opposite_mini ...the composite function is fg(x), (f ∘ g)(x) or f(g(x))..
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This page was written by Stephen Clarke.

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