June 2019 Paper 2 Q10
10
(a) \(\mathrm{f}(x) = x^3 - 2\)
The domain of \(\mathrm{f}(x)\) is \(\quad x \leqslant 3\)
Work out the range of \(\mathrm{f}(x)\). [2 marks]
(b) \(\mathrm{g}(x) = 5 - x^2\)
The domain of \(\mathrm{g}(x)\) is \(\quad -2 \leqslant x \leqslant 1\)
Work out the range of \(\mathrm{g}(x)\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\mathrm{f}(x) \leqslant 25\) or \(25 \geqslant \mathrm{f}(x)\) | B2 | B1 \(\mathrm{f}(x) \lt 25\) or \(k \leqslant \mathrm{f}(x) \leqslant 25\) or \(k \lt \mathrm{f}(x) \leqslant 25\) where \(k\) is any number \(\lt 25\) SC1 \(\leqslant 25\) or \(x \leqslant 25\) |
Additional guidance
| Condone \(\mathrm{f}(x)\) replaced by eg \(y\) or f or f\(x\) or F(\(x\)) or F or F\(x\) or \(x^3 - 2\) in B2 or B1 responses | |
| Equivalent inequalities may be seen \(25 \gt \mathrm{f}(x)\) | B1 |
| Allow \(-\infty \lt \mathrm{f}(x) \leqslant 25\) | B2 |
| Condone \(-\infty \leqslant \mathrm{f}(x) \leqslant 25\) | B2 |
| \(-\infty \lt \mathrm{f}(x) \lt 25\) or \(-\infty \leqslant \mathrm{f}(x) \lt 25\) | B1 |
| \([-\infty,\ 25]\) or \((-\infty,\ 25]\) | B1 |
| \((-\infty,\ 25)\) | B0 |
| Condone \(\mathrm{f}(x) = \ \leqslant 25\) | B2 |
| Condone \(\mathrm{f}(x) = \ \lt 25\) | B1 |
| Condone \(\mathrm{f}(x) = x \leqslant 25\) | SC1 |
| \(\mathrm{f}(x) \leqslant 25\) in working with list of integers on answer line | B1 |
| Only a list of integers | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(1 \leqslant \mathrm{g}(x) \leqslant 5\) or \(5 \geqslant \mathrm{g}(x) \geqslant 1\) | B2 | B1 \(1 \leqslant \mathrm{g}(x) \lt 5\) or \(1 \lt \mathrm{g}(x) \leqslant 5\) or \(1 \lt \mathrm{g}(x) \lt 5\) or \(\mathrm{g}(x) \geqslant 1\) and \(\mathrm{g}(x) \leqslant 5\) or \(1 \leqslant \mathrm{g}(x) \leqslant k\) where \(k\) is a constant \(\gt 1\) or \(p \leqslant \mathrm{g}(x) \leqslant 5\) where \(p\) is a constant \(\lt 5\) SC1 \(1 \leqslant x \leqslant 5\) |
Additional guidance
| Condone \(\mathrm{g}(x)\) replaced by eg \(y\) or g or g\(x\) or f(\(x\)) or f or f\(x\) or \(5 - x^2\) in B2 or B1 responses | |
| Equivalent inequalities may be seen eg \(\ 5 \geqslant \mathrm{g}(x) \gt 1\) | B1 |
| Only \(\mathrm{g}(x) \geqslant 1\) given as the answer | B0 |
| Only \(\mathrm{g}(x) \leqslant 5\) given as the answer | B0 |
| \(1 \leqslant \mathrm{g}(x) \leqslant 4\) | B1 |
| \(1 \leqslant \mathrm{g}(x) \lt 4\) | B0 |
| \(0 \leqslant \mathrm{g}(x) \leqslant 5\) | B1 |
| \(0 \lt \mathrm{g}(x) \leqslant 5\) | B0 |
| Invalid statements do not score eg1 \(\ 1 \leqslant \mathrm{g}(x) \geqslant 5\) eg2 \(\ 1 \geqslant \mathrm{g}(x) \leqslant 5\) eg3 \(\ 6 \leqslant \mathrm{g}(x) \leqslant 5\) | B0 B0 B0 |
| \([1,\ 5]\) | B1 |
| \([1,\ 5)\) or \((1,\ 5]\) or \((1,\ 5)\) or \(1 - 5\) or \(5 - 1\) | B0 |
| \(1 \leqslant \mathrm{g}(x) \leqslant 5\) in working with list of integers on answer line | B1 |
| Only a list of integers | B0 |