O-LevelAdditional MathematicsFunctions4037/21 · May/June 2023 · Q811 Marks
The diagram shows the graph of y = f(x) where f is defined by f(x) = 3x/√(5x+1) for 0 ≤ x ≤ 3.
- (a)(i)Given that f is a one-one function, find the domain and range of f⁻¹.[3]
- (a)(ii)Solve the equation f(x) = x.[2]
- (a)(iii)On the diagram above, sketch the graph of y = f⁻¹(x).[2]
- (b)(i)The functions g and h are defined by g(x) = ³√(8x³ + 3) for x ≥ 1, h(x) = e⁴ˣ for x ≥ k. Find an expression for g⁻¹(x).[2]
- (b)(ii)State the least value of the constant k such that gh(x) can be formed.[1]
- (b)(iii)Find and simplify an expression for gh(x).[1]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- [B2] [Domain f⁻¹] 0 < x < 2.25 oe
- [B1] [Range f⁻¹ ] 0 < f⁻¹ < 3
- [2] x = 1.6 oe or x = 0
- [B1] for attempt at correct graph of inverse function drawn over correct domain soi
- [B1] for correct shape with intersection in approximately correct location
- [M1] For a complete method to find the inverse, including changing the subject and swapping the variables
- [A1] [g⁻¹(x)=] ³√(x³-3)/8 oe mark final answer
- [1] [k=] 0
- [1] ³√(8e^(12x) + 3) mark final answer
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