4400 Higher June 2005 Paper 4 Q13
13 \(\mathrm{f}\) and \(\mathrm{g}\) are functions.
\(\mathrm{f} : x \mapsto 2x - 3\)
\(\mathrm{g} : x \mapsto 1 + \sqrt{x}\)
(a) Calculate \(\mathrm{f}(-4)\) (2)
(b) Given that \(\mathrm{f}(a) = 5\), find the value of \(a\). (2)
(c) Calculate \(\mathrm{gf}(6)\) (2)
(d) Which values of \(x\) cannot be included in the domain of \(\mathrm{g}\)? (1)
(e) Find the inverse function \(\mathrm{g}^{-1}\) in the form \(\mathrm{g}^{-1} : x \mapsto \ldots\) (3)
| Scheme | Marks |
|---|---|
| \(2(-4) - 3\) | M1 |
| −11 | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(2a - 3 = 5\) or \((5 + 3)/2\) | M1 |
| 4 | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\sqrt{(2 \times 6 - 3)} + 1\) | M1 |
| 4 | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Negative or \(x < 0\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(y = 1 + \sqrt{x}\) \(x = (y - 1)^2\) \(\quad\) (\(\sqrt{\phantom{x}}\), +1 becomes −1, \((\ )^2\)) | M1 M1 |
| \(\mathrm{g}^{-1} : x \to (x - 1)^2\) or \(y = (x - 1)^2\) | A1 |
| (3) | |
| (10 marks) |
Notes
M1: Or \(x = 1 + \sqrt{y}\)
A1: Or \(\mathrm{g}^{-1}(x) = (x - 1)^2\) or \((x - 1)^2\)