O-LevelAdditional MathematicsFunctions4037/22 · May/June 2018 · Q107 Marks

On the axes below, sketch the graph of y = |(x + 3)(x-5)| showing the coordinates of the points where the curve meets the x-axis. y X

(a)(i)On the axes below, sketch the graph of y = |(x + 3)(x-5)| showing the coordinates of the points where the curve meets the x-axis. y X[2]
(a)(ii)Write down a suitable domain for the function f(x) = |(x + 3)(x-5)| such that f has an inverse.[1]
(b)The functions g and h are defined by g(x) = 3x-1 for x > 1, h(x) = 4/x for x ≠ 0.
(b)(i)Find hg(x).[1]
(b)(ii)Find (hg)⁻¹(x).[2]
(c)Given that p(a) = b and that the function p has an inverse, write down p⁻¹(b).[1]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[B1] for correct shape (parabola opening upwards, reflected portion above x-axis)
[B1] for roots marked on the graph (x = -3, x = 5) or seen nearby provided graph drawn and one root is negative and one is positive
[B1] Any correct domain (e.g., x ≥ 5, x ≤ -3, or x between vertex and a root like x ≥ 1, x ≤ 1)
[B1] 4/(3x-1)
[M1] Correct method for finding inverse function e.g. swopping variables and changing subject or vice versa; or indicates (hg)⁻¹(x) = g⁻¹h⁻¹(x) and finds g⁻¹(x) = (x+1)/3 and h⁻¹(x) = 4/x
[A1] (hg)⁻¹(x) = (4+x)/(3x) oe isw (FT only if their hg(x) of the form a/(bx+c) where a, b and c are integers)
[B1] a cao

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