| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a standard C3 inverse function question with routine steps: finding an inverse by swapping x and y, stating domain/range, sketching, and using a given iterative formula. All techniques are straightforward applications of core methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(y = \ln(4-2x)\); \(e^y = 4-2x\) leading to \(x = 2 - \frac{1}{2}e^y\); Changing subject and removing ln; \(y = 2 - \frac{1}{2}e^x\) | M1, A1, A1 | cso (4 marks) |
| (b) Range of \(f^{-1}\) is \(f^{-1}(x) < 2\) (and \(f^{-1}(x) \in \mathbb{R}\)) | B1 | (1 mark) |
| (c) Graph showing: Shape, horizontal asymptote at \(y=2\), curve passing through approximately \((0, 1.5)\) and \((\ln 4, 0)\) | B1, B1, B1, B1 | (4 marks) |
| (d) \(x_1 \approx -0.3704\), \(x_2 \approx -0.3452\) | B1, B1 | cao (2 marks); Guidance: If more than 4 dp given in this part a maximum of one mark is lost. Penalise on the first occasion. |
| (e) \(x_1 = -0.354030 19\ldots\); \(x_2 = -0.350926 88\ldots\); \(x_3 = -0.35201761\ldots\); \(x_4 = -0.35163386\ldots\); \(k \approx -0.352\); Calculating to at least \(x_0\) to at least four dp | M1, A1 | cao (2 marks) |
| Alternative to (e): Let \(g(x) = x + \frac{1}{2}e^x\); \(g(-0.3515) \approx +0.0003\), \(g(-0.3525) \approx -0.001\); Change of sign (and continuity) \(\Rightarrow k \in (-0.3525, -0.3515) \Rightarrow k = -0.352\) (to 3 dp); Found in any way | M1, A1 | (2 marks) |
(a) $y = \ln(4-2x)$; $e^y = 4-2x$ leading to $x = 2 - \frac{1}{2}e^y$; Changing subject and removing ln; $y = 2 - \frac{1}{2}e^x$ | M1, A1, A1 | cso (4 marks)
(b) Range of $f^{-1}$ is $f^{-1}(x) < 2$ (and $f^{-1}(x) \in \mathbb{R}$) | B1 | (1 mark)
(c) Graph showing: Shape, horizontal asymptote at $y=2$, curve passing through approximately $(0, 1.5)$ and $(\ln 4, 0)$ | B1, B1, B1, B1 | (4 marks)
(d) $x_1 \approx -0.3704$, $x_2 \approx -0.3452$ | B1, B1 | cao (2 marks); **Guidance:** If more than 4 dp given in this part a maximum of one mark is lost. Penalise on the first occasion.
(e) $x_1 = -0.354030 19\ldots$; $x_2 = -0.350926 88\ldots$; $x_3 = -0.35201761\ldots$; $x_4 = -0.35163386\ldots$; $k \approx -0.352$; Calculating to at least $x_0$ to at least four dp | M1, A1 | cao (2 marks)
**Alternative to (e):** Let $g(x) = x + \frac{1}{2}e^x$; $g(-0.3515) \approx +0.0003$, $g(-0.3525) \approx -0.001$; Change of sign (and continuity) $\Rightarrow k \in (-0.3525, -0.3515) \Rightarrow k = -0.352$ (to 3 dp); Found in any way | M1, A1 | (2 marks)
---\begin{enumerate}
\item The function $f$ is defined by
\end{enumerate}
$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$
(a) Show that the inverse function of f is defined by
$$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$
and write down the domain of $\mathrm { f } ^ { - 1 }$.\\
(b) Write down the range of $\mathrm { f } ^ { - 1 }$.\\
(c) In the space provided on page 16, sketch the graph of $y = f ^ { - 1 } ( x )$. State the coordinates of the points of intersection with the $x$ and $y$ axes.
The graph of $y = x + 2$ crosses the graph of $y = f ^ { - 1 } ( x )$ at $x = k$.
The iterative formula
$$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$
is used to find an approximate value for $k$.\\
(d) Calculate the values of $x _ { 1 }$ and $x _ { 2 }$, giving your answers to 4 decimal places.\\
(e) Find the value of $k$ to 3 decimal places.
\hfill \mbox{\textit{Edexcel C3 2007 Q6 [13]}}