June 2023 Paper 1 Q1
1 The function f is given by \(\quad \mathrm{f}(x) = 2x + 1\)
(a) Work out \(x\) when \(\quad \mathrm{f}(x) = -5\) [2 marks]
(b) The function g is given by \(\quad \mathrm{g}(x) = x^2\)
Work out \(\quad \mathrm{fg}(3)\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(2x + 1 = -5\) or \(2x = -6\) or \(\mathrm{f}^{-1}(x) = \dfrac{x - 1}{2}\) | M1 | |
| \(x = -3\) | A1 |
Additional guidance
| Correct answer not from incorrect working | M1 A1 |
| \(\mathrm{f}(x) = 5\) followed by \(x = 2\) | SC1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| (g(3) =) 9 | B1 | may be implied by 19 if no other working |
| 19 or f [their g(3)] correctly evaluated | B1ft | any incorrect value for g(3) used correctly in f. Must be evidence of g to gain this mark (can’t be f(anything)) |
| Alternative method 2 | ||
| (\(\mathrm{fg}(x)\) =) \(2x^2 + 1\) or \(2 \times 3^2 + 1\) | B1 | may be implied by 19 if no other working |
| 19 | B1ft | |
Additional guidance
| 49 from \(\mathrm{gf}(x)\) on answer line | SC1 |
| Common errors from Alt 2 will come from getting \(\mathrm{fg}(x)\) incorrect and might include (but must include the function clearly stated): \(\mathrm{fg}(x) = 4x^2 + 1\) followed by \(\mathrm{fg}(3) = 37\) \(\mathrm{fg}(x) = x^2(2x + 1)\) followed by \(\mathrm{fg}(3) = 63\) | B0 B1ft |
| \(\mathrm{ff}(x)\) doesn’t get any marks | B0 B0 |
| Common errors from Alt 1 will come from getting \(\mathrm{g}(x)\) incorrect and might include g(3) = 6 then fg(3) = 13 g(3) = 3 then fg(3) = 7 | B0 B1ft |