C3 January 2006 Q8
8. The functions f and g are defined by\[\begin{aligned}&\mathrm{f} : x \rightarrow 2x + \ln 2, &&x \in \mathbb{R},\\&\mathrm{g} : x \rightarrow \mathrm{e}^{2x}, &&x \in \mathbb{R}.\end{aligned}\]
(a) Prove that the composite function gf is\[\mathrm{gf} : x \rightarrow 4\mathrm{e}^{4x}, \quad x \in \mathbb{R}.\] (4)
(b) In the space provided, sketch the curve with equation \(y = \mathrm{gf}(x)\), and show the coordinates of the point where the curve cuts the \(y\)-axis. (1)
(c) Write down the range of gf. (1)
(d) Find the value of \(x\) for which \(\dfrac{\mathrm{d}}{\mathrm{d}x}[\mathrm{gf}(x)] = 3\), giving your answer to 3 significant figures. (4)
| Scheme | Marks |
|---|---|
| \(\mathrm{gf}(x) = \mathrm{e}^{2(2x + \ln 2)}\) | M1 |
| \(= \mathrm{e}^{4x}\mathrm{e}^{2\ln 2}\) | M1 |
| \(= \mathrm{e}^{4x}\mathrm{e}^{\ln 4}\) | M1 |
| \(= 4\mathrm{e}^{4x}\) Give mark at this point, cso | A1 |
| \(\left(\text{Hence } \mathrm{gf} : x \mapsto 4\mathrm{e}^{4x},\ x \in \mathbb{R}\right)\) | |
| (4) |
| Scheme | Marks |
|---|---|
| Shape and point | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Range is \(\mathbb{R}_+\) Accept \(\mathrm{gf}(x) \gt 0,\ y \gt 0\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}}{\mathrm{d}x}[\mathrm{gf}(x)] = 16\mathrm{e}^{4x}\) | |
| \(\mathrm{e}^{4x} = \dfrac{3}{16}\) | M1 A1 |
| \(4x = \ln\dfrac{3}{16}\) | M1 |
| \(x \approx -0.418\) | A1 |
| (4) | |
| (10 marks) |