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Finding a Composite Function
(KS4, Year 10)
The Lesson
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) = x^{2} + 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
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) = x^{2} + 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) = x^{2} + 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) = x^{2} + 4x + 4 + 1 | Use the FOIL method to expand the brackets |
fg(x) = x^{2} + 4x + 4 + 1 | Collect the constant terms |
fg(x) = x^{2} + 4x + 5 |
Answer:
The composite function fg(x) of f(x) = x^{2} + 1 and g(x) = x + 2 is: fg(x) = x^{2} + 4x + 5.A Note on Notation
The image below shows a composite function gf(x), where a function f is passed to a function g:The composite function is gf(x).
- f is applied to the input x.
- g is applied to the function f.
...the composite function is fg(x), (f ∘ g)(x) or f(g(x))..
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