Edexcel IALMathematicsGeneral Applications7 Marks
The function f is defined by f(x) = 3x² - 6x + 5, x ∈ ℝ, x ≥ 1.
- (a)Find the range of f.[3]
- (b)Find f^{-1}(x), stating its domain.[4]
Mark scheme
Marking points from the official mark scheme. The mark or mark type is shown in brackets.
- (a) M1: for attempting to complete the square or find the vertex.
- f(x) = 3(x^2 - 2x) + 5 = 3((x-1)^2 - 1) + 5 = 3(x-1)^2 - 3 + 5 = 3(x-1)^2 + 2
- A1: for correctly completing the square or identifying vertex (1, 2).
- A1: for stating range f(x) ≥ 2 (or y ≥ 2 or [2, ∞)).
- (b) M1: for setting y = f(x) and attempting to make x the subject.
- y = 3(x-1)² + 2 ⇒ y-2 = 3(x-1)² ⇒ (y-2)/3 = (x-1)²
- x-1 = ±√((y-2)/3)
- M1: for choosing the positive root (as x ≥ 1, so x-1 ≥ 0).
- x = 1 + √((y-2)/3)
- A1: f⁻¹(x) = 1 + √((x-2)/3) (must be in terms of x).
- B1: for stating domain of f⁻¹(x) is x ≥ 2.
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