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Year 11
Higher

Solving equations involving composite functions

I can solve equations involving composite functions.

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New
New
Year 11
Higher

Solving equations involving composite functions

I can solve equations involving composite functions.

Download all resources
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Share activities with pupils
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Lesson details

Key learning points

  1. A composite function may have a given output.
  2. It is possible to solve to find the input.
  3. 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 (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.
$${\text{fg}}(x)$$, $${\text{fgh}}(x)$$, 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: $$7$$
$${\sqrt{50}}$$

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: $$x=5$$