Higher November 2017 Paper 1 Q21
21. \(f(x) = 5x + 2\)
\(g(x) = x^3\)
(a) Solve \(f^{-1}(x) = 10\). [4]
(b)
(i) Show that \(gf(x) = 125x^3 + 150x^2 + 60x + 8\). [3]
(ii) Find \(gf(-1)\). [1]
| Scheme | Marks | Comments |
|---|---|---|
| \(f^{-1}(x) = \dfrac{x - 2}{5}\) or equivalent | B2 | Award B1 for \(x = \dfrac{y - 2}{5}\) or equivalent unless \(x\) and \(y\) interchanged later or SC1 for \(y\) or \(f^{-1}(x) = \dfrac{x + 2}{5}\) or equivalent |
| \(\dfrac{x - 2}{5} = 10\) | M1 | |
| \(x = 52\) | A1 |
Alternative method
| Scheme | Marks |
|---|---|
| \(f^{-1}(x) = 10\) means \(x = f(10)\) | B2 |
| \(f(10) = 5(10) + 2\) | M1 |
| \(x = 52\) | A1 |
| Scheme | Marks | Comments |
|---|---|---|
| (i) \(gf(x) = g(5x + 2)\) or \(gf(x) = (5x + 2)^3\) or \(gf(x) = (f(x))^3\) | B1 | Correct order of composition seen or implied. |
| \((5x + 2)(5x + 2)^2 = (5x + 2)(25x^2 + 20x + 4)\) | M1 | Seen or implied. Allow \((5x + 2)^2 = 25x^2 + 20x + 4\) if \((5x + 2)^3\) attempted |
| Convincing correct completion to given answer \(125x^3 + 150x^2 + 60x + 8\) | A1 | NB Answer is given |
| (ii) \(-27\) | B1 | |
| (8) |