O-LevelAdditional MathematicsFunctions4037/22 · May/June 2021 · Q137 Marks

The functions f and g are defined, for x > 0, by f(x) = (2x²-1)/(3x) and g(x) = 1/x.

(a)Find and simplify an expression for fg(x).[2]
(b)(i)Given that f⁻¹ exists, write down the range of f⁻¹.[1]
(b)(ii)Show that f⁻¹(x) = (px + √qx² + r) / 4, where p, q and r are integers.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

(a) M1 [fg(x) =] (2(1/x)²-1)/(3(1/x)) oe
(a) A1 mark final answer [fg(x) =] (2-x²)/(3x) or 2/(3x) - x/3
(b)(i) B1 f⁻¹ > 0
(b)(ii) B1 2x²-3xy-1=0 or 2y²-3xy-1=0
(b)(ii) M1 FT their 2x² -3xy-1=0 or 2y² -3xy-1=0 with at most one sign error in the equation Correctly applies quadratic formula: [x =] -(-3y) ± √((-3y)² - 4(2)(-1))/(2(2)) oe or [y =] -(-3x) ± √((3x)² - 4(2)(-1))/(2(2)) oe
(b)(ii) B1 Justifies the positive square root at some point
(b)(ii) A1 must be a function of x f⁻¹(x) = (3x + √(9x² + 8))/4 cao

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Topic: Functions, O-Level Additional Mathematics (syllabus 4037). Structured question, 7 marks.
Source: O-Level Additional Mathematics, May/June 2021 Paper 2 Variant 2, question 13.

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