| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Sketch function and inverse graphs |
| Difficulty | Moderate -0.3 This is a straightforward composite/inverse functions question requiring completing the square (routine), identifying range/domain (direct consequence), sketching three related graphs (standard technique with y=x as line of symmetry), and finding an inverse algebraically (standard procedure for quadratics). All parts are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02m Graphs of functions: difference between plotting and sketching1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2(x-2)^2 + 2\) | B1, B1, B1 | [3] For \(2, -2, 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2 \leq f(x) \leq 10\) oe | B1 | [1] Allow \(<\) etc. Ignore notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2 \leq x \leq 10\) | B1\(\sqrt{}\) | [1] Ft from part (ii). Ignore notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x)\): approx half parabola from \((0,10)\) to \((2,2)\) | B1 | Or from int with \(y\) axis to int with *their* \(y = x\) |
| \(g(x)\): line through \(0\) at \(\approx 45°\) | B1 | |
| \(f^{-1}(x)\): reflection of *their* \(f(x)\) in \(g(x)\) | B1\(\sqrt{}\) | |
| Everything totally correct | B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x-2)^2 = \frac{1}{2}(y-2)\) | M1 | Allow \(+\sqrt{}\) or \(-\sqrt{}\). Dep on final ans as \(f^{-1}\) of \(x\) |
| \(x = 2 \pm \sqrt{\frac{1}{2}(y-2)}\) | M1 | |
| \(f^{-1}(x) = 2 - \sqrt{\frac{1}{2}(x-2)}\) | A1 [3] | cao |
## Question 11:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2(x-2)^2 + 2$ | B1, B1, B1 | [3] For $2, -2, 2$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2 \leq f(x) \leq 10$ oe | B1 | [1] Allow $<$ etc. Ignore notation |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2 \leq x \leq 10$ | B1$\sqrt{}$ | [1] Ft from part **(ii)**. Ignore notation |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x)$: approx half parabola from $(0,10)$ to $(2,2)$ | B1 | Or from int with $y$ axis to int with *their* $y = x$ |
| $g(x)$: line through $0$ at $\approx 45°$ | B1 | |
| $f^{-1}(x)$: reflection of *their* $f(x)$ in $g(x)$ | B1$\sqrt{}$ | |
| Everything totally correct | B1 | [4] |
## Question (v):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-2)^2 = \frac{1}{2}(y-2)$ | M1 | Allow $+\sqrt{}$ or $-\sqrt{}$. Dep on final ans as $f^{-1}$ of $x$ |
| $x = 2 \pm \sqrt{\frac{1}{2}(y-2)}$ | M1 | |
| $f^{-1}(x) = 2 - \sqrt{\frac{1}{2}(x-2)}$ | A1 [3] | cao |11 Functions $f$ and $g$ are defined by
$$\begin{array} { l l }
\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 10 & \text { for } 0 \leqslant x \leqslant 2 \\
\mathrm {~g} : x \mapsto x & \text { for } 0 \leqslant x \leqslant 10
\end{array}$$
(i) Express $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.\\
(ii) State the range of f .\\
(iii) State the domain of $\mathrm { f } ^ { - 1 }$.\\
(iv) Sketch on the same diagram the graphs of $y = \mathrm { f } ( x ) , y = \mathrm { g } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$, making clear the relationship between the graphs.\\
(v) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2011 Q11 [12]}}