| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Piecewise function inverses |
| Difficulty | Standard +0.3 This is a straightforward piecewise function question requiring students to find the range, sketch the inverse by reflection, and find inverse expressions for two simple functions (quadratic and linear). While it involves multiple parts, each step uses standard techniques taught in P1 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02u 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 |
| Range is \(0 \leq f(x) \leq 4\), 0 to 4 | B1 | Accept in two parts; Condone \(<\) |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = x\) drawn or implied | B1 | |
| Correct sketch of \(f^{-1}\) | B1 | SC if f missing, \((2,2)\ (4,6)\) must be shown |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((x \mapsto)\sqrt{2x}\) for \(0 \leq x \leq 2\) | B1B1 | Condone \(<\ <\) |
| \((x \mapsto)2x - 2\) for \(2 < x \leq 4\) | B1B1 | |
| [4] |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Range is $0 \leq f(x) \leq 4$, 0 to 4 | B1 | Accept in two parts; Condone $<$ |
| **[1]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = x$ drawn or implied | B1 | |
| Correct sketch of $f^{-1}$ | B1 | SC if f missing, $(2,2)\ (4,6)$ must be shown |
| **[2]** | | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x \mapsto)\sqrt{2x}$ for $0 \leq x \leq 2$ | B1B1 | Condone $<\ <$ |
| $(x \mapsto)2x - 2$ for $2 < x \leq 4$ | B1B1 | |
| **[4]** | | |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_778_816_255_662}
The diagram shows the function f defined for $0 \leqslant x \leqslant 6$ by
$$\begin{aligned}
& x \mapsto \frac { 1 } { 2 } x ^ { 2 } \quad \text { for } 0 \leqslant x \leqslant 2 , \\
& x \mapsto \frac { 1 } { 2 } x + 1 \text { for } 2 < x \leqslant 6 .
\end{aligned}$$
(i) State the range of f .\\
(ii) Copy the diagram and on your copy sketch the graph of $y = \mathrm { f } ^ { - 1 } ( x )$.\\
(iii) Obtain expressions to define $\mathrm { f } ^ { - 1 } ( x )$, giving the set of values of $x$ for which each expression is valid.
\hfill \mbox{\textit{CAIE P1 2010 Q7 [7]}}