4MA0 Higher January 2018 Paper 3 Q23
23 f is the function such that \(\mathrm{f}(x) = \dfrac{3x}{x - 2}\) where \(x \neq 2\)
g is the function such that \(\mathrm{g}(x) = \dfrac{4x}{5}\)
(a) Find \(\mathrm{gf}(-4)\) (2)
(b) Express the composite function fg in the form \(\mathrm{fg}(x) = \ldots\)
Give your answer as a single fraction in its simplest form. (3)
Give your answer as a single fraction in its simplest form. (3)
(c) Express the inverse function \(\mathrm{f}^{-1}\) in the form \(\mathrm{f}^{-1}(x) = \ldots\) (3)
| Scheme | Marks |
|---|---|
\((\mathrm{f}(-4) =)\;\dfrac{3 \times -4}{-4 - 2}\;(= 2)\) or \((\mathrm{gf}(x) =)\;\dfrac{4\left(\dfrac{3x}{x - 2}\right)}{5}\) oe or \(\dfrac{4\left(\dfrac{3 \times -4}{-4 - 2}\right)}{5}\) | M1 |
| 1.6 | A1 |
| (2) |
Notes
A1: oe
| Scheme | Marks |
|---|---|
| \(\dfrac{3 \times \dfrac{4x}{5}}{\dfrac{4x}{5} - 2}\) | M1 |
| E.g. \(\dfrac{12x}{4x - 10}\) oe | M1 |
| \(\dfrac{6x}{2x - 5}\) | A1 |
| (3) |
Notes
M1: for a correct first expression for \(\mathrm{fg}(x)\)
M1: for a correct unsimplified fraction of the form \(\dfrac{ax}{bx - c}\) where \(a\), \(b\) and \(c\) are integers
A1: cao
SC : Award B2 for \(\dfrac{3x}{x - 2.5}\)
| Scheme | Marks |
|---|---|
\(y = \dfrac{3x}{x - 2}\) or \(x = \dfrac{3y}{y - 2}\) \(y(x - 2) = 3x\) or \(x(y - 2) = 3y\) | M1 |
| \(xy - 3x = 2y\) or \(x(y - 3) = 2y\) or \(xy - 3y = 2x\) or \(y(x - 3) = 2x\) | M1 |
| \(\dfrac{2x}{x - 3}\) | A1 |
| (3) |
Notes
M1: for writing function in the form \(y = \dfrac{3x}{x - 2}\) or \(x = \dfrac{3y}{y - 2}\) and multiplying both sides by the denominator
M1: isolating terms in \(x\) (or \(y\))
A1: oe must be in terms of \(x\)