June 2025 Paper 1 Q1
1
(a)
The function f is given by \(\quad \mathrm{f}(x) = 5x + 3\)
The range is \(\quad \mathrm{f}(x) \lt 23\)
Work out the domain of the function, giving your answer as an inequality. [1 mark]
(b)
The function g is given by \(\quad \mathrm{g}(x) = x^3 - 4\)
The domain is \(\quad -3 \lt x \lt 2\)
Work out the range of the function, giving your answer as an inequality. [2 marks]
(c)
The function h is given by \(\quad \mathrm{h}(x) = \sqrt{2x + 1}\)
Work out \(\mathrm{h}^{-1}(x)\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(x \lt 4\) or \(4 \gt x\) | B1 |
Additional guidance
| Incorrect inequality symbol eg \(x \leqslant 4\) | B0 |
| \(\mathrm{f}(x) \lt 4\) or just \(\lt 4\) or just 4 (even if correct answer seen in working) | B0 |
| \(=\) used in working but correct answer | B1 |
| \(a \lt x \lt 4\) | B0 |
| \(x \leqslant 3\) | B0 |
| \(\{x : x \lt 4\}\) | B1 |
| Nothing on answer line then mark final line of working | |
| B1 response with a list of integers on answer line | B0 |
| Only a list of integers | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(-31 \lt \mathrm{g}(x) \lt 4\) | B2 | oe eg \(4 \gt \mathrm{g}(x) \gt -31\) word ‘and’ or ‘\(\cap\)’ must be included if writing two inequalities for B2 or B1 eg \(-31 \lt \mathrm{g}(x)\) and \(\mathrm{g}(x) \lt 4\) B1 \(k \lt \mathrm{g}(x) \lt 4\) where \(k\) is less than 4 or \(k \leqslant \mathrm{g}(x) \lt 4\) where \(k\) is less than 4 or \(-31 \lt \mathrm{g}(x) \lt m\) where \(m\) is greater than \(-31\) or \(-31 \lt \mathrm{g}(x) \leqslant m\) where \(m\) is greater than \(-31\) or \(-31\) and 4 seen from correct working (but within parameters of the Additional Guidance notes about listing integers) |
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 \(x^3 - 4\) eg1 \(-31 \lt \mathrm{f}(x) \lt 4\) eg2 \(-31 \lt y \lt 4\) | B2 |
| Condone \(\mathrm{g}(x)\) replaced by eg \(y\) or g or g\(x\) or f or f\(x\) or G or G\(x\) or \(x^3 - 4\) for any of the B1 answers. eg1 \(k \lt \mathrm{f}(x) \lt 4\) where \(k\) is less than 4 eg2 \(-31 \lt y \lt m\) where \(m\) is greater than \(-31\) | B1 |
| Do not condone \(\mathrm{g}(x)\) replaced by \(x\) but could still score B1 for \(-31\) and 4 seen from correct working | |
| \((-31, 4)\) or \(\{\mathrm{g}(x) : -31 \lt \mathrm{g}(x) \lt 4\}\) | B2 |
| \([-31, 4]\) or \((-31, 4]\) or \([-31, 4)\) | B1 |
| Condone eg \(\mathrm{g}(x) = -31 \lt \mathrm{g}(x) \lt 4\) | B2 |
| Just one inequality on answer line without seeing the other value in the working eg \(-31 \lt \mathrm{g}(x)\) | B0 |
| B2 response with a list of integers on answer line | B1 |
| B1 response with a list of integers on answer line or just a list of integers | B0 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: reversing the terms then rearranging | ||
| \(x = \sqrt{\left(2\mathrm{h}^{-1}(x) + 1\right)}\) or \(x = \sqrt{(2y + 1)}\) | M1 | oe |
| \(\dfrac{x^2 - 1}{2}\) | A1 | oe eg \(\dfrac{x^2}{2} - \dfrac{1}{2}\) |
| Alternative method 2: rearranging then reversing the terms | ||
| \((x =)\ \dfrac{\mathrm{h}(x)^2 - 1}{2}\) or \((x =)\ \dfrac{y^2 - 1}{2}\) | M1 | oe |
| \(\dfrac{x^2 - 1}{2}\) | A1 | oe eg \(\dfrac{x^2}{2} - \dfrac{1}{2}\) |
Additional guidance
| Answer left as \(y = \dfrac{x^2 - 1}{2}\) | M1A0 |
| Using a flow chart with incorrect answer | M0A0 |
| Using a flow chart with correct answer | M1A1 |