November 2021 Paper 1 Q4
4
(a) The function f is given by \(\quad \mathrm{f}(x) = 3x - 5\)
The range is \(\quad 13 \lt \mathrm{f}(x) \lt 19\)
Work out the domain of the function. [1 mark]
(b) The function g is given by \(\quad \mathrm{g}(x) = x^2 - 4 \quad\) with domain \(\quad -1 \lt x \lt 3\)
Work out the range of the function. [2 marks]
(c) The function h is given by \(\quad \mathrm{h}(x) = \dfrac{3 + x}{2}\)
Work out \(\mathrm{h}^{-1}(x)\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(6 \lt x \lt 8\) or \(x \lt 8\) and \(x \gt 6\) | B1 | oe eg \(8 \gt x \gt 6\) word ‘and’ must be included if writing two inequalities |
Additional guidance
| [6, 8] | B1 |
| Condone eg \(x =\ 6 \lt x \lt 8\) | B1 |
| Incorrect inequality symbol eg \(6 \leqslant x \lt 8\) or \(6 \lt x \gt 8\) | B0 |
| \(6 \lt \mathrm{f}(x) \lt 8\) | B0 |
| B1 response with a list of integers on answer line | B0 |
| Only a list of integers | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(-4 \leqslant \mathrm{g}(x) \lt 5\) or \(\mathrm{g}(x) \lt 5\) and \(\mathrm{g}(x) \geqslant -4\) | B2 | oe eg \(5 \gt \mathrm{g}(x) \geqslant -4\) word ‘and’ must be included if writing two inequalities for B2 or B1 or SC1 B1 \(-4 \lt \mathrm{g}(x) \lt 5\) or \(-4 \lt \mathrm{g}(x) \leqslant 5\) or \(-4 \leqslant \mathrm{g}(x) \leqslant 5\) or \(\mathrm{g}(x) \lt 5\) and \(\mathrm{g}(x) \gt -4\) or \(\mathrm{g}(x) \leqslant 5\) and \(\mathrm{g}(x) \gt -4\) or \(\mathrm{g}(x) \leqslant 5\) and \(\mathrm{g}(x) \geqslant -4\) or \(\mathrm{k} \lt \mathrm{g}(x) \lt 5\) where k is less than 5 or \(\mathrm{k} \leqslant \mathrm{g}(x) \lt 5\) where k is less than 5 or \(-4 \leqslant \mathrm{g}(x) \lt \mathrm{m}\) where m is greater than \(-4\) SC1 \(-4 \leqslant x \lt 5\) or \(x \lt 5\) and \(x \geqslant -4\) or only \(-4\) and 5 seen (condone 9 given as a range in this case) |
Additional guidance
| Condone \(\mathrm{g}(x)\) replaced by eg \(y\) or g or g\(x\) or f or f\(x\) or G or G\(x\) or \(x^2 - 4\) eg1 \(-4 \leqslant \mathrm{f}(x) \lt 5\) eg2 \(-4 \leqslant y \leqslant 5\) | B2 B1 |
| \([-4, 5)\) | B2 |
| \((-4, 5)\) or \((-4, 5]\) or \([-4, 5]\) | B1 |
| Condone eg \(\mathrm{g}(x) = -4 \leqslant \mathrm{g}(x) \lt 5\) | B2 |
| Condone eg \(\mathrm{g}(x) = -4 \lt \mathrm{g}(x) \lt 5\) | B1 |
| B2 response with a list of integers on answer line | B1 |
| B1 response with a list of integers on answer line | B0 |
| Only a list of integers | B0 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(x = 2\mathrm{h}(x) - 3\) or \(x = 2y - 3\) | M1 | oe |
| \(2x - 3\) | A1 | |
| Alternative method 2 | ||
| \(x = \dfrac{3 + h^{-1}(x)}{2}\) or \(x = \dfrac{3 + y}{2}\) | M1 | oe |
| \(2x - 3\) | A1 | |
Additional guidance
| Answer left as \(y = 2x - 3\) | M1A0 |