The functions f and g are defined for all real values of x by f(x) = (3x-2)²+k and g(x) = 5x-1, where k is a constant.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [*M1] Attempt to form expression for gf(x) Expect 5((3x-2)²+k)-1; fg(x) is M0. Do not allow algebraic errors.
2. [A1] Obtain 5(3x-2)²+5k−1 OE e.g. 45x² – 60x + 5k +19 .
3. [DM1] Their 5k-1 = 39 or 5k-1 ≥ 39 Or use b² – 4ac = 0 (must be '= 0', could be implied later) on 45x² −60x+5k+19-39≥0 OE.
4. [A1] Obtain k =8 Do not accept k≥8.
5. [M1] Obtaining (3(5x-1)-2)² + their k May simplify and/or use k at this stage; k may have come from an inequality in (a).
6. [A1 FT] Conclude [fg(x)]≥8 allow [y]≥8 Following their value of k; must be ≥, not >. Allow an accurate written description.
7. [B1] State g¯¹(x)=⅕(x+1) OE ⅕(x+1) must be indicated as the inverse.
8. [B1 B1] [h(x)=]7x+4 If 7x+4 only, it must be clear that this is h(x).