| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Composite transformation sketch |
| Difficulty | Moderate -0.3 This is a straightforward piecewise function question requiring standard techniques: reading range from a graph, function composition with simple substitution, solving linear and exponential equations, and stating a basic inverse function criterion. All parts are routine C3 exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties |
| Answer | Marks |
|---|---|
| (a) \(f(x) \geq -3\) | B1 |
| (b) \(f(0) = 5\) or attempts to put their \(f(0)\) into \(e^{2x - 8} - 4\) | M1 |
| Correct answer \(f(0) = e^2 - 4\) | A1 |
| (c) Either \(5 - 2x = 21 \Rightarrow x = -8\) | M1, A1 |
| Or \(e^{2x - 8} - 4 = 21\) | M1 |
| Correct order and \(\ln\) work \(\Rightarrow x = \frac{\ln 25 + 8}{2}\) or \(x = \frac{\ln 5 + 4}{2}\) or equivalent | M1, A1 |
| (d) \(f\) does not have an inverse as it is a 'many to one' function | B1 |
| Accept: \(f\) does not have an inverse as it is not a 'one to one' function |
(a) $f(x) \geq -3$ | B1
(b) $f(0) = 5$ or attempts to put their $f(0)$ into $e^{2x - 8} - 4$ | M1
Correct answer $f(0) = e^2 - 4$ | A1
(c) Either $5 - 2x = 21 \Rightarrow x = -8$ | M1, A1
Or $e^{2x - 8} - 4 = 21$ | M1
Correct order and $\ln$ work $\Rightarrow x = \frac{\ln 25 + 8}{2}$ or $x = \frac{\ln 5 + 4}{2}$ or equivalent | M1, A1
(d) $f$ does not have an inverse as it is a 'many to one' function | B1
Accept: $f$ does not have an inverse as it is not a 'one to one' function |
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-10_775_1392_233_278}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of the graph of $y = \mathrm { f } ( x )$ where
$$\mathrm { f } ( x ) = \left\{ \begin{array} { r r }
5 - 2 x , & x \leqslant 4 \\
\mathrm { e } ^ { 2 x - 8 } - 4 , & x > 4
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item State the range of $\mathrm { f } ( x )$.
\item Determine the exact value of ff(0).
\item Solve $\mathrm { f } ( x ) = 21$
Give each answer as an exact answer.
\item Explain why the function f does not have an inverse.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2013 Q6 [9]}}