| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | March |
| Marks | 9 |
| Topic | Composite & Inverse Functions |
| Type | State domain or range |
| Difficulty | Moderate -0.3 This is a straightforward inverse function question requiring standard techniques: identifying domain/range by swapping, sketching by reflecting in y=x, and using the derivative relationship (f^{-1})'(a) = 1/f'(f^{-1}(a)). While it requires multiple steps and careful attention to the restricted domain, all techniques are routine A-level procedures with no novel problem-solving required. Slightly easier than average due to the simple quadratic function. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07m Tangents and normals: gradient and equations |
$$\text{f}(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$
\begin{enumerate}[label=(\alph*)]
\item Write down the domain and range of $\text{f}^{-1}$ [2]
\item Sketch the graph of $\text{f}^{-1}$, indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
\item Find the gradient of $f^{-1}(x)$ when $f^{-1}(x) = \frac{5}{3}$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q3 [9]}}