| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | October |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Standard +0.3 This is a straightforward composition of functions question requiring basic algebraic manipulation. Part (a) involves simple substitution to find fg(x) and gf(x), while part (b) requires solving a polynomial equation that factors nicely. The question is slightly easier than average as it's mostly procedural with no conceptual challenges beyond understanding function composition. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
In this question you must show detailed reasoning.
The functions f and g are defined for all real values of $x$ by
$$f(x) = x^3 \text{ and } g(x) = x^2 + 2.$$
\begin{enumerate}[label=(\alph*)]
\item Write down expressions for
\begin{enumerate}[label=(\roman*)]
\item $fg(x)$, [1]
\item $gf(x)$. [1]
\end{enumerate}
\item Hence find the values of $x$ for which $fg(x) - gf(x) = 24$. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q6 [8]}}