| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Moderate -0.8 Part (a) is a routine inverse function calculation requiring algebraic manipulation (squaring and rearranging) with straightforward domain identification. Part (b) tests basic understanding that many-to-one functions have no inverse. Both parts are standard textbook exercises with no problem-solving or novel insight required, making this 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 | Mark | Guidance |
| \(y = \sqrt{1+2x} \Rightarrow y^2 = 1 + 2x\) | M1 | Starting to work through inverse processes. Allow equivalent processes e.g. swapping \(x\) and \(y\) at other stages |
| \(\left[\text{f}^{-1}(x) =\right] \frac{x^2-1}{2}\) oe | A1 | |
| The domain is \(x \geq 0\) | B1 | B0 if \(y \geq 0\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| It is many-to-one or Is not one-to-one or It is one-to-many | E1 | Could be via an example e.g. \(x=2\) and \(x=-2\) give the same value of \(g(x)\). Give BOD for use of 'it' provided it does not contradict e.g. \(g(x)\) is one-to-many does not score. isw after a correct answer |
| [1] |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = \sqrt{1+2x} \Rightarrow y^2 = 1 + 2x$ | M1 | Starting to work through inverse processes. Allow equivalent processes e.g. swapping $x$ and $y$ at other stages |
| $\left[\text{f}^{-1}(x) =\right] \frac{x^2-1}{2}$ oe | A1 | |
| The domain is $x \geq 0$ | B1 | **B0** if $y \geq 0$ |
| **[3]** | | |
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## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| It is many-to-one **or** Is not one-to-one **or** It is one-to-many | E1 | Could be via an example e.g. $x=2$ and $x=-2$ give the same value of $g(x)$. Give BOD for use of 'it' provided it does not contradict e.g. $g(x)$ is one-to-many does not score. isw after a correct answer |
| **[1]** | | |
---2
\begin{enumerate}[label=(\alph*)]
\item The function $\mathrm { f } ( x )$ is defined by
$$f ( x ) = \sqrt { 1 + 2 x } \text { for } x \geqslant - \frac { 1 } { 2 }$$
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of this inverse function.
\item Explain why $\mathrm { g } ( x ) = 1 + x ^ { 2 }$, with domain all real numbers, has no inverse function.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2024 Q2 [4]}}