A-LevelMathematicsFunctions9709/12 · Feb/Mar 2021 · Q78 Marks
Functions f and g are defined as follows: f : x → x² + 2x + 3 for x ≤ −1, g: x → 2x + 1 for x ≥ −1.
- (a)Express f(x) in the form (x + a)² + b and state the range of f.[3]
- (b)Find an expression for f⁻¹(x).[2]
- (c)Solve the equation gf(x) = 13.[3]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.
- [2] [f(x) =](x+1)² + 2 (Accept a = 1,b = 2.)
- [1] Range [of f is (y)] ≥ 2 (OE. Do not allow x ≥ 2, FT on their b.)
- [1] y = (x+1)² + 2 leading to x = [±]√y-2-1 (Or by using the formula. Allow one sign error.)
- [1] f⁻¹(x) = -√x−2-1
- [1] 2(x²+2x+3)+1=13 (Or using a correct completed square form of f(x).)
- [1] 2x²+4x-6[=0] leading to (2)(x-1)(x+3)[=0] (Or x = 1, x = −3 using formula or completing square. Must reach 2 solutions.)
- [1] x = -3 only
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