O-LevelAdditional MathematicsFunctions4037/22 · May/June 2017 · Q99 Marks

A function f is defined, for x ≤ 3/2, by f(x) = 2x² - 6x + 5.

(i)Express f(x) in the form a(x – b)² + c, where a, b and c are constants.[3]
(ii)On the same axes, sketch the graphs of y = f(x) and y = f⁻¹(x), showing the geometrical relationship between them. [Figure 9.1][3]
(iii)Using your answer from part (i), find an expression for f⁻¹(x), stating its domain.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[B3] 2(x-1.5)² + 0.5 isw
[B3] for a = 2 and b = 1.5 and c = 0.5 provided not from wrong format isw
[B2] for 2(x-1.5)² + c where c ≠ 0.5 or a = 2 and b = 1.5
[SC2] for 2(x-1.5) + 0.5 or 2((x-1.5)²+1/4) seen
[B1] for (x-1.5)² seen or for b = 1.5 or for c = 0.5
[SC1] for 3 correct values seen in incorrect format e.g. 2(x-1.5x)+0.5 or 2(x²-1.5)+0.5
[B3] [B1] for correct graph for f over correct domain or correct graph for f⁻¹ over correct domain
[B1] for vertex marked for f or f⁻¹ and intercept marked for f or f⁻¹
[B1] for idea of symmetry – either symmetrical by eye, ignoring any scale or line y = x drawn and labelled
Maximum of 2 marks if not fully correct
[M1] FT their a,b,c, provided their a ≠1 and a,b,c are all non-zero constants or y-0.5/2 = (x-1.5)² and reverses variables at some point
[A1] f⁻¹(x)=1.5 - √( (x-0.5)/2 ) oe must have selected negative square root only; condone y = etc.; must be in terms of x
[B1] x ≥ 0.5 oe
[SC2] for f⁻¹(x) = (6-√(8x-4))/4 oe
[SC1] for f'(x) = (-(-6)±√(36-4(2)(5-x)))/2(2) oe

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