Higher June 2019 Paper 1 Q19
19. The functions \(f(x)\) and \(g(x)\) are defined for \(x \gt 0\) by
\[\begin{aligned} f(x) &= \frac{8}{x}\,,\\ g(x) &= x + 5. \end{aligned}\](a) Find and simplify an expression for \(ff(x)\). [2]
(b) Using your answer to part (a), or otherwise, explain the relationship between \(f(x)\) and \(f^{-1}(x)\). [1]
(c) Solve \(g^{-1}f(x) = 11\). [4]
| Scheme | Marks | Comments |
|---|---|---|
| \(8 \div \dfrac{8}{x}\) oe | M1 | |
| \(x\) | A1 |
| Scheme | Marks | Comments |
|---|---|---|
| Valid explanation e.g. ‘They are the same function.’ | E1 | allow e.g. ‘\(f(x)\) is a self-inverse function’; allow the inverse function found and the explanation stated |
| Scheme | Marks | Comments |
|---|---|---|
| \(\left(g^{-1}(x) =\right)\ x - 5\) | B1 | |
| \(\left(g^{-1}f(x) =\right)\ \dfrac{8}{x} - 5\) | B1 | |
| \(\dfrac{8}{x} = 16\) or better | M1 | |
| \((x =)\ \dfrac12\) oe | A1 | CAO |
Alternative method
| Scheme | Marks | Comments |
|---|---|---|
| \(f(x) = g(11)\) | M1 | |
| \(f(x) = 16\) | M1 | |
| \(\dfrac{8}{x} = 16\) or better | M1 | |
| \((x =)\ \dfrac12\) oe | A1 |
| (7) |