The functions f and fg are defined by f(x) = e^(x²+3) for x < 0 fg(x) = e^(2x) for x > 3/2

Mark scheme

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

1. [B1] Valid explanation using f: f is one-one oe
2. [M1] Complete method to find inverse function: Swaps the variables and rearranges or rearranges and swaps the variables
3. [A2] [f⁻¹(x)=] -√(lnx−3) isw or [f⁻¹(x)=] -√(ln(x/e³)) oe isw
4. [A2] A1 for [f⁻¹(x) =] [±]√(lnx−3) or A1 for [f⁻¹(x) =] [±]√(ln(x/e³)) oe
5. [B1] Domain f⁻¹: x > e³
6. [B1] Range f⁻¹: f⁻¹ < 0
7. [M1] g(x) = f⁻¹(e^(2x)) soi or g(x) = -√(ln(e^(2x)) – 3)
8. [M1] FT their expression for f⁻¹
9. [A1] -√(2x-3)
10. [A1] If 0 scored, allow SCB1 for -√(2x-3) found from solving (g(x))² + 3 = 2x and using existence of composite functions to deduce that the square root must be negative