| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic function concepts. Part (a) requires recognizing that ± gives two outputs (not a function by definition), part (b) is routine composition requiring algebraic substitution, and part (c) follows directly from the domain restriction. All parts are standard textbook exercises with no problem-solving or novel insight required. |
| 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 | Marks | Guidance |
| because it's neither a one-to-one nor a many-to-one (mapping) | B1 | allow because it's one-to-many (mapping); allow eg because each value of \(x\) is mapped to two values oe; do not allow eg because it's a one-to-many function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x^2+2}{x^2+2-3}\) | M1 | |
| \(\frac{x^2+2}{x^2-1}\) or \(\frac{x^2+2}{(x-1)(x+1)}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | x | > 1\) or \(x < -1\) or \(x > 1\) or \(x < -1, x > 1\) or \(x < -1 \cup x > 1\) |
## Question 12(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| because it's neither a one-to-one nor a many-to-one (mapping) | B1 | allow because it's one-to-many (mapping); allow eg because each value of $x$ is mapped to two values oe; do not allow eg because it's a one-to-many **function** |
---
## Question 12(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x^2+2}{x^2+2-3}$ | M1 | |
| $\frac{x^2+2}{x^2-1}$ or $\frac{x^2+2}{(x-1)(x+1)}$ | A1 | |
---
## Question 12(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|x| > 1$ or $x < -1$ or $x > 1$ or $x < -1, x > 1$ or $x < -1 \cup x > 1$ | B1 | do not allow eg $x < -1$ and $x > 1$; eg $-1 > x > 1$ |
---12 It is given that
\begin{itemize}
\item $\mathrm { f } ( x ) = \pm \frac { 1 } { \sqrt { x } } , x > 0$
\item $\mathrm { g } ( x ) = \frac { x } { x - 3 } , x > 3$
\item $\mathrm { h } ( x ) = x ^ { 2 } + 2 , x \in \mathbb { R }$.
\begin{enumerate}[label=(\alph*)]
\item Explain why $\mathrm { f } ( x )$ is not a function.
\item Find $\mathrm { gh } ( x )$.
\item State the domain of $\mathrm { gh } ( x )$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2023 Q12 [4]}}