| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a multi-part question on composite and inverse functions with standard techniques. Parts (a)(i) and (a)(ii) involve routine substitution and range finding for a rational function. Part (a)(iii) requires the standard algebraic manipulation to find an inverse (swap x and y, rearrange). Part (b) uses the given condition to find k by substituting into the inverse function—straightforward algebra. While it has multiple parts (typical of 4+ marks), each step uses well-practiced A-level techniques without requiring novel insight or particularly complex manipulation. Slightly easier than average due to the structured, procedural nature. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
4.
The functions $f$ and $g$ are defined by
$$\begin{array} { l l l }
\mathrm { f } ( x ) = \frac { k x } { 2 x - 1 } & x \in \mathbb { R } & x \neq \frac { 1 } { 2 } \\
\mathrm {~g} ( x ) = 2 + 3 x - x ^ { 2 } & x \in \mathbb { R } &
\end{array}$$
where $k$ is a non-zero constant.
\begin{enumerate}[label=(\alph*)]
\item Find in terms of $k$
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { fg } ( 4 )$
\item the range of f
\item $\mathrm { f } ^ { - 1 }$
Given that
$$\mathrm { f } ^ { - 1 } ( 2 ) = \frac { 11 } { 3 \mathrm {~g} ( 2 ) }$$
\end{enumerate}\item find the exact value of $k$\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q4 [8]}}