O-LevelAdditional MathematicsFunctions4037/21 · May/June 2014 · Q1212 Marks

The functions f and g are defined by f(x) = 2x/(x+1) for x > 0, g(x) = √(x+1) for x > -1.

(i)Find fg(8).[2]
(ii)Find an expression for f²(x), giving your answer in the form ax/(bx+c), where a, b and c are integers to be found.[3]
(iii)Find an expression for g⁻¹(x), stating its domain and range.[4]
(iv)On the same axes, sketch the graphs of y = g(x) and y = g⁻¹(x), indicating the geometrical relationship between the graphs.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

[M1] f(3) or fg(x) = 2√(x+1) / (√(x+1)+1)
[A1] 6/4 or oe
[M1] allow omission of 2(.....) in numerator or (.....) + 1 in denominator, but not both.
[dM1] A correct and valid step in simplification e.g. multiplying numerator and denominator by x + 1,, or simplifying 2x/(x+1) / (2x/(x+1)+1) to (2x + x + 1)/(x + 1)
[A1] 4x / (3x +1) Correctly simplified to 4x/(3x+1)
[M1] Putting y = g(x), condone x = y² -1; reasonable changing subject to x and swopping x and y or vice versa attempt at correct method
[A1] g⁻¹(x) = x² -1 condone y =....., f' =......
[B1] (Domain) x > 0 condone y >-1 f'>-1
[B1] (Range) g(x) > −1
[B1+B1] correct graphs; -1 need not be labelled but could be implied by 'one square'
[B1] idea of reflection or symmetry in line y = x must be stated.

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