A-LevelMathematicsFunctions9709/12 · Feb/Mar 2020 · Q98 Marks

The function f is defined by f(x) = 2x² + 12x + 11 for x ≤ −4.

(a)Express 2x² + 12x + 11 in the form 2(x + a)² + b, where a and b are constants.[2]
(b)Find an expression for f⁻¹(x) and state the domain of f⁻¹.[3]
(c)For the case where k = −1, solve the equation fg(x) = 193.[2]
(d)State the largest value of k possible for the composition fg to be defined.[1]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.

[B1B1] (a) [2(x+3)²] [-7] Stating a=3,b=-7 gets B1B1
[M1] (b) y = 2(x+3)² -7 → 2(x+3)² = y +7 → (x+3)² = (y+7)/2 First 2 operations correct. Condone sign error or with x/y interchange
[A1FT] (b) x+3=(±)√((y+7)/2) → x=(±)√((y+7)/2) -3 → f⁻¹(x)= -√((x+7)/2) -3 FT on their a and b. Allow y = ...
[B1] (b) Domain: x ≥ −5 or ≥ -5 or [-5, ∞) Do not accept y = ..., f (x)=..., f⁻¹ (x) = ...
[B1FT] (c) fg(x) = 8x² -7 SOI. FT on their –7 from part (a)
[B1] (c) 8x²-7=193 → x² = 25 → x = -5 only
[B1] (d) (Largest k is) -1/2 Accept -1/2 or k < -1/2

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Topic: Functions, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, Feb/Mar 2020 Paper 1 Variant 2, question 9.

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