The function f is defined by f(x) = −3x² + 2 for x ≤ −1.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

1. [B1] [y]≤-1 Accept for f(x)<-1, -∞<y<−1, (-∞,-1]. Do not accept x≤−1.
2. [M1] y = −3x² +2 rearranged to 3x² = 2−y, leading to x² = (2-y)/3 or y² = (2-x)/3
3. [A1 A1] x = -√( (2-y)/3 ) → [f⁻¹(x)] = -√( (2-x)/3 ) Al for minus, Al for √( (2-x)/3 ), allow √( (x-2)/-3 )
4. [M1] fg(x) = -3(-x²-1)² +2 SOI expect -3x⁴-6x²-1.
5. [M1] gf(x) = -(-3x²+2)² -1 SOI expect –9x⁴ +12x² -5.
6. [A1] fg(x)- gf(x)+8=0 leading to 6x⁴ -18x² +12 [=0] OE
7. [M1] [6](x²-1)(x²-2) [=0] or formula or completion of the square Solving a 3-term quadratic equation in x² must be seen.
8. [A1] x=-1,-√2 only these two solutions Allow -1, -1.41[4] Answers only SC B1.