| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Standard +0.2 This is a straightforward multi-part question testing standard C3 techniques: proving a function is even (simple substitution), differentiation using chain rule, understanding inverses and their domains, and verifying the inverse function derivative relationship. All parts are routine textbook exercises requiring no novel insight, making it easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions |
The function $f(x) = \ln(1 + x^2)$ has domain $-3 \leq x \leq 3$.
Fig. 9 shows the graph of $y = f(x)$.
\includegraphics{figure_9}
\begin{enumerate}[label=(\roman*)]
\item Show algebraically that the function is even. State how this property relates to the shape of the curve. [3]
\item Find the gradient of the curve at the point P$(2, \ln 5)$. [4]
\item Explain why the function does not have an inverse for the domain $-3 \leq x \leq 3$. [1]
\end{enumerate}
The domain of $f(x)$ is now restricted to $0 \leq x \leq 3$. The inverse of $f(x)$ is the function $g(x)$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Sketch the curves $y = f(x)$ and $y = g(x)$ on the same axes.
State the domain of the function $g(x)$.
Show that $g(x) = \sqrt{e^x - 1}$. [6]
\item Differentiate $g(x)$. Hence verify that $g'(\ln 5) = \frac{1}{4}$. Explain the connection between this result and your answer to part (ii). [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q9 [19]}}