| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.8 This is a straightforward composite function question requiring basic substitution, domain/range identification from exponential properties, and solving a simple equation by working backwards through the composition. All techniques are routine A-level procedures with no novel problem-solving required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties |
$$f(x) = e^x, x \in \mathbb{R}, x > 0.$$
$$g(x) = 2x^3 + 11, x \in \mathbb{R}.$$
\begin{enumerate}[label=(\alph*)]
\item Find and simplify an expression for the composite function $gf(x)$. [2]
\item State the domain and range of $gf(x)$. [2]
\item Solve the equation
$$gf(x) = 27.$$ [3]
\end{enumerate}
The equation $gf(x) = k$, where $k$ is a constant, has solutions.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item State the range of the possible values of $k$. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q4 [8]}}