C3 June 2005 Q3
3. The function f is defined by\[\mathrm{f} : x \rightarrow \frac{5x + 1}{x^2 + x - 2} - \frac{3}{x + 2}, \quad x \gt 1.\]
(a) Show that \(\mathrm{f}(x) = \dfrac{2}{x - 1}\), \(x \gt 1\). (4)
(b) Find \(\mathrm{f}^{-1}(x)\). (3)
The function g is defined by\[\mathrm{g} : x \rightarrow x^2 + 5, \quad x \in \mathbb{R}.\]
(c) Solve \(\mathrm{fg}(x) = \tfrac{1}{4}\). (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{5x + 1}{(x + 2)(x - 1)} - \dfrac{3}{x + 2}\) | B1 |
| \(= \dfrac{5x + 1 - 3(x - 1)}{(x + 2)(x - 1)}\) M1 for combining fractions even if the denominator is not lowest common | M1 |
| \(= \dfrac{2x + 4}{(x + 2)(x - 1)} = \dfrac{2(x + 2)}{(x + 2)(x - 1)} = \dfrac{2}{x - 1}\) * M1 must have linear numerator | M1 A1 cso |
| (4) |
| Scheme | Marks |
|---|---|
| \(y = \dfrac{2}{x - 1} \Rightarrow xy - y = 2 \Rightarrow xy = 2 + y\) | M1 A1 |
| \(\mathrm{f}^{-1}(x) = \dfrac{2 + x}{x}\) o.e. | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathrm{fg}(x) = \dfrac{2}{x^2 + 4}\) (attempt) \(\left[\dfrac{2}{\text{"}g\text{"} - 1}\right]\) | M1 |
| Setting \(\dfrac{2}{x^2 + 4} = \dfrac{1}{4}\) and finding \(x^2 = \ldots\); \(\quad x = \pm 2\) | M1; A1 |
| (3) | |
| (10 marks) |