4MA0 Higher January 2014 Paper 3R Q22
22 The functions f and g are such that \(\;\mathrm{f}(x) = x + 3\;\) and \(\;\mathrm{g}(x) = \dfrac{1}{x - 2}\)
(a) Find \(\mathrm{fg}(x)\)
Give your answer as a single algebraic fraction expressed as simply as possible. (3)
Give your answer as a single algebraic fraction expressed as simply as possible. (3)
(b) Express the inverse function \(\mathrm{g}^{-1}\) in the form \(\mathrm{g}^{-1}(x) = \ldots\) (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{x - 2} + 3\) | M1 |
| \(\dfrac{1 + 3(x - 2)}{x - 2}\) or \(\dfrac{1 + 3x - 6}{x - 2}\) | M1 |
| \(\dfrac{3x - 5}{x - 2}\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(y(x - 2) = 1\) or \(xy - 2y + 1\) | M1 |
| \(xy = 2y + 1\) | M1 |
| \(\dfrac{2x + 1}{x}\) | A1 |
| (3) | |
| (6 marks) |
Notes
A1: or \(\dfrac{1}{x} + 2\)
Alternative method
\(x(y - 2) = 1\) or \(xy - 2x = 1\)
\(xy = 1 + 2x\)
(corrected from the printed mark scheme: the question total is printed as “Total 5 marks”; the parts are worth 3 + 3 = 6 marks, as on the question paper)