The functions f and g are defined by f(x) = (2x+3)/(x-1), x ∈ ℝ, x ≠ 1 g(x) = 2x² - 3, x ∈ ℝ (a) Find the range of f. (b) Find f⁻¹(x). (c) Solve the equation fg(x) = 1/2.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. (a) M1 for attempting to divide numerator by denominator, or considering the horizontal asymptote by looking at coefficients of x.
2. (a) A1 for y ∈ ℝ, y ≠ 2 (accept f(x) ≠ 2)
3. (b) M1 for setting y = f(x) and attempting to make x the subject, e.g., y(x-1) = 2x+3
4. (b) A1 for yx - y = 2x + 3
5. (b) M1 for collecting x terms and factorising, x(y-2) = y+3
6. (b) A1 for f⁻¹(x) = (x+3)/(x-2) (accept with different variable, e.g., y)
7. (c) M1 for attempting to find fg(x) by substituting g(x) into f(x)
8. (c) A1 for fg(x) = (2(2x²-3)+3)/((2x²-3)-1) = (4x²-6+3)/(2x²-4) = (4x²-3)/(2x²-4)
9. (c) M1 for setting fg(x) = 1/2 and attempting to solve for x (e.g., cross-multiplication)
10. (c) A1 for 2(4x²-3) = 2x²-4 ⇒ 8x²-6 = 2x²-4 ⇒ 6x² = 2 ⇒ x² = 1/3
11. (c) A1 for x = ± 1/√3 or x = ± √3/3