The functions f and g are defined for x ≥ 0 by f : x → 2x² + 3, g: x → 3x + 2.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2017). The mark or mark type is shown in brackets.

1. [B1] gf(x) = 3(2x² + 3) + 2 = 6x² + 11 AG
2. [B1] fg(x) = 2(3x+2)² + 3 Allow 18x² + 24x + 11 ISW if simplified incorrectly. Not retrospectively from (ii)
3. [M1] y = 2(3x+2)² + 3 => 3x+2 = (±)sqrt((y-3)/2) oe Subtract 3;divide by 2;square root. Or x/y interchanged. Allow sqrt(y-3)/2 for 1st M
4. [M1] => x = (±)sqrt((y-3)/2) - 2/3 oe Subtract 2; divide by 3; Indep. of 1st M1. Or x/y interchanged.
5. [A1] (fg)⁻¹(x) = (1/3)(sqrt((x-3)/2) - 2) oe Must be a function of x. Allow alt. method g⁻¹f⁻¹ (x)
6. [M1] Solve their (fg)⁻¹(x) ≥ 0 or attempt range of fg
7. [A1] Domain is x ≥ 11
8. [M1] Replace x with 2x in gf and equate to their fg(x) from (i). Allow 12x² + 11 =
9. [A1] Collect terms to obtain correct quadratic expression. 6x² - 24x = 0 oe
10. [A1] Both required x=0, 4