Grade 12 Maths · Functions

Find the equation of an inverse

Find the inverse of a function by swapping x and y, then making y the subject (for example y = 2x gives y = x ÷ 2).

past paper questions use it: 16 past paper questions use it
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8) Inverse equations | Many examplesKevinmathscience

2) Inverse of straight lineKevinmathscience

3) Inverse of parabola | Part 1Kevinmathscience

6) Inverse of exponential | Part 1Kevinmathscience

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Question 6.4 · May/June 2025 · Paper 1 · National (DBE) · 2 marks

The graphs of

f (x) = p^{x} + q

and

g (x) = m x + c

are drawn below.

A (3; 4)

is the point of intersection of

f

and

g

B (0; - 3)

is the y-intercept of

f

. E is the x-intercept of

g

6.4 Write down the equation of

g^{- 1}

in the form

y = \dots

(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: find the equation of an inverse

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3. Question 8.3.3 · November 2024 · Paper 1 · National (DBE) · 3 marks
8.3 Given: $f (x) = - 6 x^{2}$
8.3.3 Determine the equation of $f^{- 1}$ for $f^{- 1} (x) \leq 0$ . Write your answer in the form $y = ...$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Swap x and y $x = - 6 y^{2}$
  1 mark · Find the equation of an inverse
2. ✓
  Solve for y $y^{2} = - \frac{x}{6}$
  $y = \pm - \frac{x}{6}$
  1 mark · Find the equation of an inverse
3. ✓
  Apply restrictionSince $y \leq 0$ :
  $y = - - \frac{x}{6}$
  1 mark · Restrict domain for an inverse
4. Question 4.4 · November 2024 · Paper 1 · National (DBE) · 3 marks
Given: $f (x) = a^{x} - 1$ for $a > 0$ . $B (2; \frac{- 5}{9})$ is a point on $f$ .
4.4 It is further given that $C$ is a point on $f$ at $y = \frac{19}{8}$ . Determine the coordinates of $C^{'}$ , the image of $C$ , when $C$ is reflected about the line $y = x$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Substitute y-value to find x $\frac{19}{8} = (\frac{2}{3})^{x} - 1$
  $\frac{27}{8} = (\frac{2}{3})^{x}$
  1 mark · Substitute into a formula
2. ✓
  Solve for x $(\frac{3}{2})^{3} = (\frac{3}{2})^{- x}$
  $3 = - x ⟹ x = - 3$
  Coordinates of C are $(- 3; \frac{19}{8})$
  1 mark · Solve an exponential equation
3. ✓
  Reflect point across y=xSwap x and y coordinates.
  $C^{'} (\frac{19}{8}; - 3)$
  1 mark · Find the equation of an inverse
5. Question 4.2 · September 2024 · Paper 1 · Provincial · 2 marks
The graph of $f (x) = a x^{2}$ , $x \leq 0$ , is sketched below. The point $P (- 6; - 8)$ lies on the graph of f.
4.2 Determine the equation of $f^{- 1}$ , in the form $y =$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Interchange x and y $x = - \frac{2}{9} y^{2}$
  1 mark · Find the equation of an inverse
2. ✓
  Answer $y = - - \frac{9}{2} x$
  1 mark · Find the equation of an inverse
6. Question 6.4.2 · March 2023 · Provincial · 2 marks
The graphs of $f (x) = a x^{2}$ and $g (x) = b^{x}$ are sketched. Points $A (- 1; 2)$ and $B$ are points of intersection.
6.4 If $h (x) = g (x + 3)$ , write down the coordinates of:
6.4.2 $A^{''}$ , the new coordinates of $A$ on the graph of $h^{- 1}$ , the inverse of $h$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Swap coordinates of A'From $A^{'} (- 4; 2)$ , swap x and y.
  $A^{''} (2; - 4)$
  2 marks · Find the equation of an inverse

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