Higher June 2023 Paper 1 Q20
20 For \(x \geqslant 0\), the functions f and g are such that
\[\mathrm{f}(x) = 3x + 4 \qquad \mathrm{g}(x) = \dfrac{\sqrt{x} + 2}{5}\](a) Find \(\mathrm{g}^{-1}(x)\) (2)
(b) Solve \(\mathrm{gf}(x) = 3\) (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| \((5x - 2)^2\) | M1 | for a first step to change the subject of \(y = \dfrac{\sqrt{x} + 2}{5}\) or \(x = \dfrac{\sqrt{y} + 2}{5}\), eg \(5y = \sqrt{x} + 2\) or \(5x = \sqrt{y} + 2\) |
| A1 | oe |
| Answer | Mark | Mark scheme |
|---|---|---|
| 55 | M1 | for method to find \(\mathrm{gf}(x)\), eg \(\dfrac{\sqrt{3x + 4} + 2}{5}\) or method to find \(\mathrm{g}^{-1}(3)\), eg \((5 \times 3 - 2)^2\ (= 169)\) |
| M1 | for method to solve as far as 2 steps away, eg \((5 \times 3 - 2)^2 = 3x + 4\) or \(3x + 4 = \text{``}169\text{''}\) or for a complete method, eg \((\text{``}169\text{''} - 4) \div 3\) | |
| A1 | cao |