| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 6 |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.8 This is a straightforward composite function question requiring basic substitution (gf(x) = √(1-2^x)), simple domain analysis (2^x ≤ 1 gives x ≤ 0), and range identification. All parts follow standard procedures with no novel problem-solving required, making it easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
10.
The functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = 2 ^ { x } , x \in \mathbb { R } \\
& \mathrm {~g} ( x ) = \sqrt { 1 - x } , x \in \mathbb { R } , x \leq a
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item State the maximum possible value of $a$.
\item The function h is defined by $\mathrm { h } ( x ) = \mathrm { gf } ( x )$\\
(b) (i) Write down an expression for $\mathrm { h } ( x )$\\
(b) (ii) Using set notation, state the greatest possible domain of h .\\
(b) (iii) State the range of h .
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q10 [6]}}