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Writing composite functions

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Lesson details

Learning outcome

I can write a composite function both numerically and algebraically.

Key learning points

  1. Using function notation, you can write a composite function.
  2. Function notation can be replaced with the algebraic form.
  3. An input can be substituted in and the numerical output found.

Keywords

  • Composite function - The composite function gf($$x$$) is the combination of f($$x$$) and g($$x$$). The output from f($$x$$) is the input for g($$x$$).

Common misconception

Pupils may try to evaluate composite functions by working from left to right.

Remind pupils that they cannot do this as each function needs an input and with composite functions the input needs to be evaluated before it can be used.

Teacher tip

In the second learning cycle you may wish to extend your pupils' learning by asking them to consider what the domain and range of the composite functions would be.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Prior knowledge starter quiz

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6 Questions

Q1.
For the function $$\text{f}(x)$$ if $$\text{f}(10)=35$$ then $$10$$ is a value in the...

expression
Correct answer: domain
range
output

Q2.
Simplify $$3x + 12 - x - 7$$.

$$2x-5$$
Correct answer: $$2x+5$$
$$4x+19$$
$$8$$
$$3x+5$$

Q3.
Simplify $$4(x-3)+x-3$$.

$$5x-6$$
Correct answer: $$5(x-3)$$
Correct answer: $$5x-15$$
$$5x+15$$
$$3x-15$$

Q4.
Expand and simplify $$(x+5)^2+(x+5)$$.

$$4x^2+100$$
Correct answer: $$x^2+11x+30$$
$$x^2+10x+25$$
$$2x^2+10x+30$$
$$2(x+5)^2$$

Q5.
Expand and simplify $$2(x+5)-(x+8)$$.

$$4$$
$$2x+18$$
$$x+18$$
Correct answer: $$x+2$$
$$2x+2$$

Q6.
If $$\text{f}(x)=5x$$, $$\text{g}(x)=3x+4$$ and $$\text{h}(x)=1-2x$$ evaluate and order the below from greatest to smallest value.

1 - $$\text{g}(3)$$
2 - $$\text{f}(2)$$
3 - $$\text{h}(-3)$$
4 - $$\text{h}(1)$$
5 - $$\text{g}(-2)$$
6 - $$\text{f}(-1)$$

Assessment exit quiz

Download exit quiz questions (PDF)

6 Questions

Q1.
__________ function is where another function is applied to the output of a function.

An algebraic
An inverse
Correct answer: A composite

Q2.
Match the notation to the correct language.

Correct Answer:$${\text{fg}}(8)$$,The output of $${\text{g}}(8)$$ being input in to $${\text{f}}(x)$$

The output of $${\text{g}}(8)$$ being input in to $${\text{f}}(x)$$

Correct Answer:$${\text{gf}}(8)$$,The output of $${\text{f}}(8)$$ being input in to $${\text{g}}(x)$$

The output of $${\text{f}}(8)$$ being input in to $${\text{g}}(x)$$

Correct Answer:$${\text{hg}}(8)$$,The output of $${\text{g}}(8)$$ being input in to $${\text{h}}(x)$$

The output of $${\text{g}}(8)$$ being input in to $${\text{h}}(x)$$

Correct Answer:$${\text{gh}}(8)$$,The output of $${\text{h}}(8)$$ being input in to $${\text{g}}(x)$$

The output of $${\text{h}}(8)$$ being input in to $${\text{g}}(x)$$

Correct Answer:$${\text{ff}}(8)$$,The output of $${\text{f}}(8)$$ being input in to $${\text{f}}(x)$$

The output of $${\text{f}}(8)$$ being input in to $${\text{f}}(x)$$

Q3.
If $${\text{f}}(x)=2x$$ and $${\text{g}}(x)=5x-1$$ then $${\text{fg}}(5)=$$ .

Correct Answer: 48, fg(5)=48

Q4.
If $${\text{f}}(x)=2x$$ and $${\text{g}}(x)=5x-1$$ then $${\text{gf}}(5)=$$ .

Correct Answer: 49, gf(5)=49

Q5.
If $${\text{f}}(x)=9x$$ and $${\text{g}}(x)=3-2x$$ how do we express the function $${\text{fg}}(x)$$?

Correct answer: $${\text{fg}}(x)=9(3-2x)$$
$${\text{fg}}(x)=3-18x$$
Correct answer: $${\text{fg}}(x)=27-18x$$
$${\text{fg}}(x)=3-9x$$
$${\text{fg}}(x)=27-2x$$

Q6.
If $${\text{f}}(x)=x^2$$ and $${\text{g}}(x)=3-2x$$, how do we express the function $${\text{fg}}(x)$$?

$${\text{fg}}(x)=3-2x^2$$
Correct answer: $${\text{fg}}(x)=(3-2x)^2$$
$${\text{fg}}(x)=3-2(x^2)$$
$${\text{fg}}(x)=9+4x^2$$
Correct answer: $${\text{fg}}(x)=9-12x+4x^2$$

To help you plan your 11 maths lesson on: Writing composite functions, download all teaching resources for free and adapt to suit your pupils' needs...