| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Challenging +1.2 This is a multi-part transformations and inverse functions question that requires understanding of function composition, one-one functions, and geometric reasoning about curve intersections. Part (i) is routine transformation identification. Part (ii) is standard inverse function work. Part (iii) requires insight about the relationship between y=f(x), y=f^{-1}(x), and y=x, plus solving an inequality—this elevates it above average difficulty but remains within typical C3 scope. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
\includegraphics{figure_9}
The function f is defined by $f(x) = \sqrt{(mx + 7)} - 4$, where $x \geq -\frac{7}{m}$ and $m$ is a positive constant. The diagram shows the curve $y = f(x)$.
\begin{enumerate}[label=(\roman*)]
\item A sequence of transformations maps the curve $y = \sqrt{x}$ to the curve $y = f(x)$. Give details of these transformations. [4]
\item Explain how you can tell that f is a one-one function and find an expression for $f^{-1}(x)$. [4]
\item It is given that the curves $y = f(x)$ and $y = f^{-1}(x)$ do not meet. Explain how it can be deduced that neither curve meets the line $y = x$, and hence determine the set of possible values of $m$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q9 [13]}}