| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | February |
| Marks | 9 |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Standard +0.3 This is a straightforward composite function question requiring substitution, completing the square to find range, solving a quadratic equation, and applying the horizontal line test. All techniques are standard A-level fare with no novel insights needed, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
For all real values of $x$, the functions $f$ and $g$ are defined by $f (x) = x^2 + 8ax + 4a^2$ and $g(x) = 6x - 2a$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find $fg(x)$. Determine the range of $fg(x)$ in terms of $a$. [4]
\item If $fg(2) = 144$, find the value of $a$. [3]
\item Determine whether the function $fg$ has an inverse. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q6 [9]}}