Functions
Inverse functions
In this note
- 2.12Understand the inverse function \(f^{-1}\) as the function that undoes \(f\).
- 2.12Find \(f^{-1}(x)\) by swapping x and y and rearranging, including fractions and the subject twice.
- 2.12Find \(f^{-1}\) of a number directly.
Before you start: Function notation and changing the subject. Next: Composite functions.
Formula bank
Undoing a function
\(f(x)=4x-3\) does × 4 then − 3. The inverse undoes these in reverse order: + 3 then ÷ 4, so \(f^{-1}(x)=\dfrac{x+3}4\). Check: \(f(5)=17\) and \(f^{-1}(17)=5\).
General methodHow to find \(f^{-1}(x)\)
- Write \(y=\) the rule.
- Swap \(x\) and \(y\).
- Rearrange to make \(y\) the subject (clear fractions; if \(y\) appears twice, collect and factorise).
- Write the answer as \(f^{-1}(x)=\ldots\), in terms of \(x\).
- Check with a number: \(f(1)\), then \(f^{-1}\) of that answer should give 1.
Example \(f(x)=\dfrac{2x+1}3\). Find \(f^{-1}(x)\).
Swap, then × 3, − 1, ÷ 2.
- \(x=\dfrac{2y+1}3\)
- \(3x=2y+1\)
- \(y=\dfrac{3x-1}2\)
Answer \(f^{-1}(x)=\dfrac{3x-1}2\)
Example 2 \(f(x)=\dfrac{3x+2}{x-1}\). Find \(f^{-1}(x)\).
Swap; \(y\) appears twice, so collect and factorise.
- \(x=\dfrac{3y+2}{y-1}\)
- \(xy-x=3y+2\)
- \(xy-3y=x+2\)
- \(y(x-3)=x+2\)
Answer \(f^{-1}(x)=\dfrac{x+2}{x-3}\)
Try it yourself
Show answer
Answer \(\dfrac{x+3}4\)
Show answer
Answer \(3(x-2)\)
Hint
Show answer
Answer \(\sqrt[3]{x-4}\)
The inverse of a number
\(f^{-1}(5)\) asks: which input gives 5 out of \(f\)? Solve \(f(x)=5\) - no formula needed. For \(f(x)=4x-3\): \(4x-3=5\Rightarrow x=2\), so \(f^{-1}(5)=2\).
Try it yourself
Show answer
Answer 2
Hint
Show answer
Answer 2
Ways they test it
Type 1 Linear inverse
"\(g(x)=5x-2\). Find \(g^{-1}(x)\)." (2 marks)
Spot it a simple rule.
- Swap.
- Undo in reverse order.
For example \(x=5y-2\)
- \(y=\dfrac{x+2}5\)
Answer \(\dfrac{x+2}5\)
Type 2 Fraction or x underneath
"\(f(x)=\dfrac5{x-2}\). Find \(f^{-1}(x)\)." (2-3 marks)
Spot it a denominator.
- Swap.
- Multiply both sides by the denominator.
- Rearrange.
For example \(x(y-2)=5\)
- \(y-2=\dfrac5x\)
Answer \(\dfrac5x+2\)
Type 3 Subject twice
"\(f(x)=\dfrac{x+4}{2x-1}\). Find \(f^{-1}(x)\)." (3 marks)
Spot it x on top AND bottom.
- Swap and clear the fraction.
- Collect the y terms; factorise y out.
- Divide.
For example \(2xy-x=y+4\)
- \(y(2x-1)=x+4\)
Answer \(\dfrac{x+4}{2x-1}\) (the same as f!)
Type 4 Inverse of a number
"Find \(f^{-1}(10)\)." (1-2 marks)
Spot it a number inside f⁻¹( ).
- Solve f(x) = 10.
Mistakes to avoid
- ✗ \(f^{-1}(x)=\dfrac1{4x-3}\)✓ \(\dfrac{x+3}4\)Inverse is not the reciprocal.
- ✗ \(\dfrac x4+3\)✓ \(\dfrac{x+3}4\)Undo in REVERSE order.
- ✗ Answer left in y✓ write it in xIt is \(f^{-1}(x)\).
Summary
- \(f^{-1}\) undoes \(f\): reverse operations in reverse order.
- Swap x and y, then make y the subject.
- \(f^{-1}(a)\): solve \(f(x)=a\).
- \(f^{-1}(x)\ne\dfrac1{f(x)}\).
If you see …
- "find f⁻¹(x)"swap x and y, rearrange.
- y on both sides after swappingcollect the y terms and factorise.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer \(\dfrac{x+2}5\)
Show answer
Answer \(\dfrac{7-x}2\)
Hint
Show answer
Answer \(8-x\)
Medium
Guide me
What is it asking? an expression in x
What do we know? swap: \(x=\tfrac{2y-1}5\)
Plan × 5, + 1, ÷ 2
Show working
- \(5x=2y-1\)
- \(y=\dfrac{5x+1}2\)
Answer \(\dfrac{5x+1}2\)
Guide me
What is it asking? an expression in x
What do we know? swap: \(x=\tfrac5{y-2}\)
Plan × \((y-2)\)
Show working
- \(x(y-2)=5\)
- \(y-2=\dfrac5x\)
- \(y=\dfrac5x+2\)
Answer \(\dfrac5x+2\)
Guide me
What is it asking? an expression in x
What do we know? swap: \(x=\tfrac1{y+3}\)
Plan \(y+3=\tfrac1x\)
Show working
- \(y+3=\dfrac1x\)
- \(y=\dfrac1x-3\)
Answer \(\dfrac1x-3\)
Hard
Guide me
What is it asking? an expression in x
What do we know? y appears twice
Plan collect and factorise
Show working
- \(x(y-1)=3y+2\)
- \(xy-3y=x+2\)
- \(y(x-3)=x+2\)
Answer \(\dfrac{x+2}{x-3}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(x(y-1)=3y+2\) | M1 | ||
| \(y(x-3)=x+2\) | M1 | ||
| \(\dfrac{x+2}{x-3}\) | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? an expression in x
What do we know? \(x(2y-1)=y+4\)
Plan \(y(2x-1)=x+4\)
Show working
- \(2xy-x=y+4\)
- \(2xy-y=x+4\)
- \(y(2x-1)=x+4\)
Answer \(\dfrac{x+4}{2x-1}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(x(2y-1)=y+4\) | M1 | ||
| \(y(2x-1)=x+4\) | M1 | ||
| \(\dfrac{x+4}{2x-1}\) | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a number
What do we know? solve \(2x^{3}-6=10\)
Plan \(x^{3}=8\)
Show working
- \(2x^{3}=16\)
- \(x^{3}=8\)
- \(x=2\)
Answer 2
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(2x^{3}-6=10\) | M1 | ||
| 2 | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I understand the inverse as the function that undoes f.
- I can find inverse functions, including fractions and the subject twice.
- I can find the inverse of a number by solving.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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