The function f : x → 6 − 4 cos(½x) is defined for 0 ≤ x ≤ 2π.

Mark scheme

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

1. [1] Set f(x) = 4: 6 − 4cos(½x) = 4 → 4cos(½x) = 2. Makes cos(½x) the subject.
2. [1] cos(½x) = ½. Looks up “½x” before ×2
3. [1] ½x = π/3 → x = 2π/3. co (120° gets A0 – decimals A0)
4. [2] The range of f is 2 ≤ f(x) ≤ 10. condone <
5. [1] The sketch graph should show a point of inflexion at x = π.
6. [1] The sketch graph should be fully correct.
7. [1] Let y = 6 − 4cos(½x). Rearrange to make cos(½x) the subject: cos(½x) = (6-y)/4. Makes cos⁻¹(½x) the subject
8. [1] Rearrange further: ½x = cos⁻¹((6-y)/4). Order of operations correct (M marks allowed if + for –)
9. [1] x = 2cos⁻¹((6-y)/4). Replace y with x for inverse function: f⁻¹(x) = 2cos⁻¹((6-x)/4). oe - needs to be a function of x not y