| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | October |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | Find range of composite |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on function composition and logarithms. Part (a) requires simple substitution: f(e²) = 3ln(e²)/2 = 3, then g(3) = 15/7. Part (b) needs finding the range of fg(x) = (3/2)ln((4x+3)/(2x+1)), which simplifies as x→∞ to approach (3/2)ln(2), giving range (0, (3/2)ln(2)). Part (c) involves recognizing f(8) = (9/2)ln(2) and f(2) = (3/2)ln(2) form a GP with ratio 1/3, then applying the sum to infinity formula. All parts use standard A-level techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.06d Natural logarithm: ln(x) function and properties |
The functions f and g are defined by
$$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$
$$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find gf($\text{e}^2$) writing your answer in simplest form. [2]
\item Find the range of the function fg. [2]
\item Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form $a \ln 2$ where $a$ is rational. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q11 [8]}}