| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.8 This is a straightforward inverse function question requiring algebraic manipulation to rearrange y = √((x+3)/2) + 1 for x in terms of y. The steps are routine: isolate the square root, square both sides, and rearrange to quadratic form. Finding the domain of f^(-1) from the range of f is standard. Easier than average as it follows a well-practiced procedure with no conceptual challenges. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = \sqrt{\frac{x+3}{2}} + 1\), for \(x \geq -3\) | ||
| Make \(x\) the subject or interchange \(x, y\) | M1 | Attempt at \(x\) as subject and removes \(+1\) |
| \(\rightarrow 2(x-1)^2 - 3\) | M1 | Squares both sides and deals with "\(+3\)" and "\(\div 2\)" |
| \(\rightarrow 2x^2 - 4x - 1\) | A1 | co |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Domain of \(f^{-1}\) is \(\geq 1\) | B1 | co, condone \(>1\) |
| [1] |
## Question 2:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = \sqrt{\frac{x+3}{2}} + 1$, for $x \geq -3$ | | |
| Make $x$ the subject or interchange $x, y$ | M1 | Attempt at $x$ as subject and removes $+1$ |
| $\rightarrow 2(x-1)^2 - 3$ | M1 | Squares both sides and deals with "$+3$" and "$\div 2$" |
| $\rightarrow 2x^2 - 4x - 1$ | A1 | co |
| **[3]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Domain of $f^{-1}$ is $\geq 1$ | B1 | co, condone $>1$ |
| **[1]** | | |
---2 A function f is such that $\mathrm { f } ( x ) = \sqrt { } \left( \frac { x + 3 } { 2 } \right) + 1$, for $x \geqslant - 3$. Find\\
(i) $\mathrm { f } ^ { - 1 } ( x )$ in the form $a x ^ { 2 } + b x + c$, where $a , b$ and $c$ are constants,\\
(ii) the domain of $\mathrm { f } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2012 Q2 [4]}}