4MA0 Higher January 2015 Paper 3 Q20
20 \(\mathrm{f}: x \mapsto 2x^2 + 1 \qquad \mathrm{g}: x \mapsto \dfrac{2x}{x - 1}\) where \(x \neq 1\)
(a) Express the composite function gf in the form \(\mathrm{gf}: x \mapsto \ldots\)
Give your answer as simply as possible. (2)
Give your answer as simply as possible. (2)
(b) Express the inverse function \(\mathrm{g}^{-1}\) in the form \(\mathrm{g}^{-1}: x \mapsto \ldots\) (3)
| Scheme | Marks |
|---|---|
| Eg \(\dfrac{2(2x^2 + 1)}{(2x^2 + 1) - 1}\) | M1 |
| \(\dfrac{(2x^2 + 1)}{x^2}\) | A1 |
| (2) |
Notes
M1: For a correct expression
A1: Accept \(2 + \dfrac{1}{x^2}\) or \(2 + x^{-2}\)
M1A0 for \(\dfrac{4x^2 + 2}{2x^2}\)
| Scheme | Marks |
|---|---|
| \(y(x - 1) = 2x\) or \(yx - y = 2x\) | M1 |
| \(x(y - 2) = y\) | M1 |
| \(\dfrac{x}{x - 2}\) | A1 |
| (3) | |
| (5 marks) |
Notes
M1: \(x(y - 1) = 2y\) or \(xy - x = 2y\)
M1: \(y(x - 2) = x\)
A1: Allow \(\dfrac{-x}{2 - x}\)