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Solving equations involving composite functions

I can solve equations involving composite functions.

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Lesson 6 of 14

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Year 11

Solving equations involving composite functions

I can solve equations involving composite functions.

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Lesson details

Key learning points

A composite function may have a given output.
It is possible to solve to find the input.
This can be extended to two composite functions that have a common output.

Keywords

Function - A function is a mathematical relationship that uniquely maps values of one set to the values of another set.
Equation - An equation is used to show two expressions that are equal to each other.

Common misconception

Confusing the order in which functions are evaluated.

Pupils can substitute their answer into the composite function to check their answer. Putting brackets around the functions can help with confirming the order.

This is considered in the second learning cycle and two methods for solving with composite functions are explored. Encourage pupils to evaluate which method they prefer and why.

Teacher tip

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).

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Prior knowledge starter quiz

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6 Questions

Q1.
$${\text{fg}}(x)$$, $${\text{fgh}}(x)$$, and $${\text{fff}}(x)$$ are all examples of __________.

quadratic equations

Correct answer: composite functions

multiple functions

Q2.
If $${\text{f}}(x)=7x−4$$ and $${\text{g}}(x)=3x$$ evaluate $${\text{gf}}(11)$$.

Correct Answer: 219, gf(11)=219

Q3.
Solve $$4(x-3)=22$$ $$x=$$ .

Correct Answer: 8.5, x=8.5

Q4.
If $${\text{f}}(x)=6x−15$$ solve $${\text{f}}(x)=57$$ $$x=$$ .

Correct Answer: 12, x = 12

Q5.
If $${\text{f}}(x)=7x−4$$ and $${\text{g}}(x)=3x$$ which is an expression for $${\text{gf}}(x)$$?

$${\text{fg}}(x)=21x-4$$

Correct answer: $${\text{fg}}(x)=3(7x-4)$$

$${\text{fg}}(x)=7(3x-4)$$

$${\text{fg}}(x)=7(3x)-4$$

Q6.
Solve $$3x^2+3=150$$.

$$-{\sqrt{50}}$$

Correct answer: $$-7$$

$$x=3$$

Correct answer: $$7$$

$${\sqrt{50}}$$

Assessment exit quiz

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6 Questions

Q1.
For the function $${\text{fg}}(x)$$ the $$x$$ values are in the domain and the values of $${\text{fg}}(x)$$ are in the __________.

Correct answer: range

domain

$$y$$ values

Q2.
If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{fg}}(x)=-3$$ $$x=$$ .

Correct Answer: -5, x=-5

Q3.
If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{gf}}(x)=-3$$ $$x=$$ .

Correct Answer: -2, x=-2

Q4.
If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{fg}}(x)={\text{gf}}(x)$$.

$$x=3$$

$$x=6$$

$$x=-6$$

$$x=-3$$

Correct answer: This equation has no solution.

Q5.
If $${\text{f}}(x)=2x-7$$ what do you notice when we solve $${\text{f}}(x)={\text{f}}^{-1}(x)$$?

$$x=0$$

$$y=x$$

Correct answer: When we input our solution into function the input and output are equal.

There is no solution.

Q6.
If $${\text{f}}(x)=7-2x$$ and $${\text{g}}(x)=x^2$$ solve $${\text{gf}}(x)=9$$.

Correct answer: $$x=2$$

$$x=-2$$

$$x=1$$

$$x=-1$$

Correct answer: $$x=5$$

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UnitMaths / Functions and proof

Maths

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Solving equations involving composite functions

Solving equations involving composite functions

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.$${\text{fg}}(x)$$, $${\text{fgh}}(x)$$, and $${\text{fff}}(x)$$ are all examples of __________.

Q2.If $${\text{f}}(x)=7x−4$$ and $${\text{g}}(x)=3x$$ evaluate $${\text{gf}}(11)$$.

Q3.Solve $$4(x-3)=22$$ $$x=$$ .

Q4.If $${\text{f}}(x)=6x−15$$ solve $${\text{f}}(x)=57$$ $$x=$$ .

Q5.If $${\text{f}}(x)=7x−4$$ and $${\text{g}}(x)=3x$$ which is an expression for $${\text{gf}}(x)$$?

Q6.Solve $$3x^2+3=150$$.

Assessment exit quiz

6 Questions

Q1.For the function $${\text{fg}}(x)$$ the $$x$$ values are in the domain and the values of $${\text{fg}}(x)$$ are in the __________.

Q2.If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{fg}}(x)=-3$$ $$x=$$ .

Q3.If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{gf}}(x)=-3$$ $$x=$$ .

Q4.If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{fg}}(x)={\text{gf}}(x)$$.

Q5.If $${\text{f}}(x)=2x-7$$ what do you notice when we solve $${\text{f}}(x)={\text{f}}^{-1}(x)$$?

Q6.If $${\text{f}}(x)=7-2x$$ and $${\text{g}}(x)=x^2$$ solve $${\text{gf}}(x)=9$$.

Lesson appears in

UnitMaths / Functions and proof

Maths

Q1.
$${\text{fg}}(x)$$, $${\text{fgh}}(x)$$, and $${\text{fff}}(x)$$ are all examples of __________.

Q2.
If $${\text{f}}(x)=7x−4$$ and $${\text{g}}(x)=3x$$ evaluate $${\text{gf}}(11)$$.

Q3.
Solve $$4(x-3)=22$$ $$x=$$ .

Q4.
If $${\text{f}}(x)=6x−15$$ solve $${\text{f}}(x)=57$$ $$x=$$ .

Q5.
If $${\text{f}}(x)=7x−4$$ and $${\text{g}}(x)=3x$$ which is an expression for $${\text{gf}}(x)$$?

Q6.
Solve $$3x^2+3=150$$.

Q1.
For the function $${\text{fg}}(x)$$ the $$x$$ values are in the domain and the values of $${\text{fg}}(x)$$ are in the __________.

Q2.
If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{fg}}(x)=-3$$ $$x=$$ .

Q3.
If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{gf}}(x)=-3$$ $$x=$$ .

Q4.
If $${\text{f}}(x)=3x$$ and $${\text{g}}(x)=2x+9$$ solve $${\text{fg}}(x)={\text{gf}}(x)$$.

Q5.
If $${\text{f}}(x)=2x-7$$ what do you notice when we solve $${\text{f}}(x)={\text{f}}^{-1}(x)$$?

Q6.
If $${\text{f}}(x)=7-2x$$ and $${\text{g}}(x)=x^2$$ solve $${\text{gf}}(x)=9$$.