Hint
Swap coordinates when reflecting. For common points, use both equations; cubing is reversible for real numbers.
Worked solution
(a)
Cubing is strictly increasing on the real line. Reverse y=x³ to obtain x=∛y, so f⁻¹(x)=∛x for every real input.
Answer: \(\sqrt[3]{x}\), domain \(\mathbb R\).
(b)
Reflection in y=x exchanges the coordinates, giving B=(8,2).
Answer: \((8,2)\); line \(y=x\).
(c)
Equate x³=∛x and cube both sides: x⁹=x. Thus x=0 or x⁸=1; for real x the latter gives x=±1. Substitution gives the three listed points, and all satisfy both original equations.
Answer: \((-1,-1),(0,0),(1,1)\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 2 marks
- 1 mark: Obtain the real cube-root inverse.
- 1 mark: State the full real domain.
Alternative methods: An equivalent real-cube-root notation is accepted.
Common error: Using a square root or excluding negative inputs.
(b) · 2 marks
- 1 mark: Give the reflected point (8,2).
- 1 mark: State the line y=x.
Alternative methods: A labelled coordinate reflection is accepted.
Common error: Changing signs instead of exchanging the coordinates.
(c) · 3 marks
- 1 mark: Use a complete method, such as the reversible equation x⁹=x.
- 1 mark: Obtain exactly x=−1,0,1.
- 1 mark: Give all three matching coordinates and check the original equations.
Alternative methods: Using strict increase to justify common points on y=x, then solving x³=x, is accepted.
Common error: Assuming a common point must be on y=x without a valid argument or losing zero by division.
Total: 7 marks.