| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation with inverses |
| Difficulty | Standard +0.3 This is a straightforward composite and inverse functions question requiring standard techniques: finding an inverse of a linear function, solving a simple equation, sketching with reflection symmetry, and solving a composite function inequality. All parts follow routine procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x + 3 = \frac{1}{2}x - \frac{3}{2} \Rightarrow x = -3\) | B1, M1A1 [3] | |
| (ii) 2 lines approximately correct, reflected in \(y=x\) & meeting at \((-3, -3)\) | B3,2,1,0 [3] | Can be implied by graph or in writing; Ignore lines extended |
| Answer | Marks | Guidance |
|---|---|---|
| \(-\frac{5}{2} \leq x \leq 0\) | M1, A1, M1, A1A1 [5] | Solving any quadratic to do with f and g \(\leq 16\), to x =; Condone < and > |
**(i)** $f^{-1}(x) = \frac{1}{2}x - \frac{3}{2}$
$2x + 3 = \frac{1}{2}x - \frac{3}{2} \Rightarrow x = -3$ | B1, M1A1 [3]
**(ii)** 2 lines approximately correct, reflected in $y=x$ & meeting at $(-3, -3)$ | B3,2,1,0 [3] | Can be implied by graph or in writing; Ignore lines extended
**(iii)** $g(x) = (2x+3)^2 - 6(2x+3)$
$4x^2 - 9$
$4x^2 - 9 \leq 16 \Rightarrow x^2 \leq \frac{25}{4}$
$-\frac{5}{2} \leq x \leq 0$ | M1, A1, M1, A1A1 [5] | Solving any quadratic to do with f and g $\leq 16$, to x =; Condone < and >
---Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by
\begin{align}
\mathrm{f} : x \mapsto 2x + 3 \quad &\text{for } x \leqslant 0,\\
\mathrm{g} : x \mapsto x^2 - 6x \quad &\text{for } x \leqslant 3.
\end{align}
\begin{enumerate}[label=(\roman*)]
\item Express $\mathrm{f}^{-1}(x)$ in terms of $x$ and solve the equation $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$. [3]
\item On the same diagram sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}^{-1}(x)$, showing the coordinates of their point of intersection and the relationship between the graphs. [3]
\item Find the set of values of $x$ which satisfy $\mathrm{gf}(x) \leqslant 16$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q9 [11]}}