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Finding the inverse of a function

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Lesson details

Learning outcome

I can find the inverse of a function.

Key learning points

  1. By changing the subject, you can find the inverse function.
  2. The inverse function maps the output to the input.
  3. There is a graphical connection between a function and its inverse.

Keywords

  • Inverse function - An inverse function reverses the mapping of the original function.

Common misconception

Confusing domain and range when considering the inverse function.

The range of a function is the domain of its inverse function.

Teacher tip

Pupils need to be able to manipulate equations in order to access this lesson. Consider reviewing these skills before this lesson if you think your pupils would benefit from this.

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.
If the vertical axis is labelled 'Range', what is the missing word on the horizontal axis?

An image in a quiz
Correct answer: Domain
Range
Input
Output
$$x$$-axis

Q2.
For the linear equation $$y=7(x-12)$$ what is the value of the $$y$$ coordinate if $$x=6$$?

Correct Answer: -42, y=-42

Q3.
For the linear equation $$y=7(x-12)$$ what is the value of the $$x$$ coordinate if $$y=35$$?

Correct Answer: 17, $$x=17$$

Q4.
Which of the below makes $$a$$ the subject of the equation $$7a+3=b$$?

Correct answer: $$a={{b-3}\over{7}}$$
$$a=7b+3$$
$$a={{b+3}\over{7}}$$
$$a={{b}\over{7}}-3$$

Q5.
Which of the below makes $$a$$ the subject of the equation $${1\over2}(5a-7)=b$$?

$${1\over2}(5b-7)=a$$
$$a={{2b-7}\over{5}}$$
Correct answer: $$a={{2b+7}\over{5}}$$
$$a={{2b}\over{5}}-7$$
$$a={{5b+7}\over{2}}$$

Q6.
Why is this function not valid?

An image in a quiz
There is a one-to-one relationship
There is a many-to-one relationship
There is a many-to-many relationship
Correct answer: There is a one-to-many relationship

Assessment exit quiz

Download exit quiz questions (PDF)

6 Questions

Q1.
$${\text{f}}(x)$$ is shorthand notation for a function of $$x$$. $${\text{f}}^{-1}(x)$$ is shorthand notation for its ...

reverse function
Correct answer: inverse function
reciprocal function

Q2.
If $${\text{f}}(x)=8x+2$$ what was the value of $$x$$ that mapped to $$34$$?

Correct Answer: 4, x=4

Q3.
Which of the below is the inverse function of $${\text{f}}(x)=7x$$?

$${\text{f}}(x)={x\over7}$$
$$-{\text{f}}(x)={x\over7}$$
Correct answer: $${\text{f}}^{-1}(x)={x\over7}$$
$${\text{f}}^{-1}(x)={7\over{x}}$$

Q4.
If $${\text{f}}(17)=391$$ then $${\text{f}}^{-1}(391)=$$ .

Correct Answer: 17

Q5.
Match the functions to their inverses.

Correct Answer:$${\text{f}}(x)=4x+5$$,$${\text{f}}^{-1}(x)={{x-5}\over{4}}$$

$${\text{f}}^{-1}(x)={{x-5}\over{4}}$$

Correct Answer:$${\text{f}}(x)=4x-5$$,$${\text{f}}^{-1}(x)={{x+5}\over{4}}$$

$${\text{f}}^{-1}(x)={{x+5}\over{4}}$$

Correct Answer:$${\text{f}}(x)=4(x+5)$$,$${\text{f}}^{-1}(x)={{x}\over{4}}-5$$

$${\text{f}}^{-1}(x)={{x}\over{4}}-5$$

Correct Answer:$${\text{f}}(x)=5x+4$$,$${\text{f}}^{-1}(x)={{x-4}\over{5}}$$

$${\text{f}}^{-1}(x)={{x-4}\over{5}}$$

Correct Answer:$${\text{f}}(x)=5(x+4)$$,$${\text{f}}^{-1}(x)={{x}\over{5}}-4$$

$${\text{f}}^{-1}(x)={{x}\over{5}}-4$$

Q6.
A function and its inverse are a reflection of one another in the line...

An image in a quiz
$$y={\text{f}}(x)$$
Correct answer: $$y=x$$
$${\text{f}}^{-1}(x)={\text{f}}(x)$$
of the $$x$$ axis
of the $$y$$ axis

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