| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 8 |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Moderate -0.8 This question involves routine operations with composite functions and absolute values. Part (c) requires straightforward evaluation: g(5) = |8-10| = 2, then f(2) = 4-6+1 = -1. The absolute value equation and range-finding are standard techniques covered in early A-level pure maths with no novel problem-solving required. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function1.02u Functions: definition and vocabulary (domain, range, mapping) |
9.
The function g is defined by
$$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph with equation $y = \mathrm { g } ( x )$, showing the coordinates of the points where the graph cuts or meets the axes.\\
(2)
\item Solve the equation
$$| 8 - 2 x | = x + 5$$
The function $f$ is defined by
$$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
\item Find fg(5).
\item Find the range of f . You must make your method clear.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q9 [8]}}