C3 June 2011 Q4
4. The function f is defined by
\[\mathrm{f} : x \mapsto 4 - \ln(x + 2), \quad x \in \mathbb{R},\ x \geqslant -1\]
(a) Find \(\mathrm{f}^{-1}(x)\). (3)
(b) Find the domain of \(\mathrm{f}^{-1}\). (1)
The function g is defined by
\[\mathrm{g} : x \mapsto \mathrm{e}^{x^2} - 2, \quad x \in \mathbb{R}\]
(c) Find \(\mathrm{fg}(x)\), giving your answer in its simplest form. (3)
(d) Find the range of fg. (1)
| Scheme | Marks |
|---|---|
| \(y = 4 - \ln(x + 2)\) | |
| \(\ln(x + 2) = 4 - y\) | |
| \(x + 2 = e^{4 - y}\) | |
| \(x = e^{4 - y} - 2\) | M1 |
| \(f^{-1}(x) = e^{4 - x} - 2\) oe | M1A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(x \leqslant 4\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(fg(x) = 4 - \ln\left(e^{x^2} - 2 + 2\right)\) | M1 |
| \(fg(x) = 4 - x^2\) | dM1A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(fg(x) \leqslant 4\) | B1ft |
| (1) | |
| (8 marks) |