| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find minimum domain for inverse |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question on standard P1 topics. Part (i) requires routine differentiation to find a stationary point. Part (ii) tests understanding that a function needs to be one-to-one for an inverse (requiring k=2 for the vertex). Parts (iii) and (iv) involve standard inverse function manipulation and sketching reflections in y=x. All techniques are textbook exercises with no novel problem-solving required, making this easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02m Graphs of functions: difference between plotting and sketching1.02v Inverse and composite functions: graphs and conditions for existence1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Stationary point at \(x = 2\); \(y\)-coordinate \(= 8\); Nature: Maximum; (or \(y = -x^2 + 4x + 4 \to -2x + 4 = 0 \to (2, 8)\) Max) | B1, B1, B1 | co; co; co independent of first two marks |
| (ii) \(k = 2\) | B1 | ✓ on "x-value" |
| (iii) \(y = 8 - (x-2)^2 \to (x-2)^2 + y = 8 \to (x-2) = \pm\sqrt{8-y} \to g^{-1} = 2 + \sqrt{8-x}\) | M1, M1, A1 | Attempt to make x the subject; Order of operations correct; Must be f(x) |
| (iv) | B1, B1, B1 | B1 arc 1st quad (no tp, no axes); B1 Evidence of symmetry about y = x; B1 all correct as shown left |
$f(x) = 8 - (x-2)^2$
**(i)** Stationary point at $x = 2$; $y$-coordinate $= 8$; Nature: Maximum; (or $y = -x^2 + 4x + 4 \to -2x + 4 = 0 \to (2, 8)$ Max) | B1, B1, B1 | co; co; co independent of first two marks
**(ii)** $k = 2$ | B1 | ✓ on "x-value"
**(iii)** $y = 8 - (x-2)^2 \to (x-2)^2 + y = 8 \to (x-2) = \pm\sqrt{8-y} \to g^{-1} = 2 + \sqrt{8-x}$ | M1, M1, A1 | Attempt to make x the subject; Order of operations correct; Must be f(x)
**(iv)** | B1, B1, B1 | B1 arc 1st quad (no tp, no axes); B1 Evidence of symmetry about y = x; B1 all correct as shown leftThe function $f$ is such that $f(x) = 8 - (x - 2)^2$, for $x \in \mathbb{R}$.
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates and the nature of the stationary point on the curve $y = f(x)$. [3]
\end{enumerate}
The function $g$ is such that $g(x) = 8 - (x - 2)^2$, for $k \leqslant x \leqslant 4$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item State the smallest value of $k$ for which $g$ has an inverse. [1]
\end{enumerate}
For this value of $k$,
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item find an expression for $g^{-1}(x)$, [3]
\item sketch, on the same diagram, the graphs of $y = g(x)$ and $y = g^{-1}(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q11 [10]}}