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Year 11
Higher

Writing composite functions

I can write a composite function both numerically and algebraically.

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New
New
Year 11
Higher

Writing composite functions

I can write a composite function both numerically and algebraically.

Download all resources
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Share activities with pupils
arrow-right
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Lesson details

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.

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.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), 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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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 - $$\text{f}(-1)$$

Exit quiz

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