| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a multi-part question on rational functions requiring identification of asymptotes from a graph, stating range, and finding an inverse function. While it involves several steps, each part uses standard techniques: reading asymptotes gives a and b, the range comes from the horizontal asymptote and domain restriction, and finding the inverse of a rational function is a routine algebraic procedure taught in C3/C4. The question is slightly easier than average because the graph provides significant scaffolding for parts (a) and (b). |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(b = 2\) | B1 | From vertical asymptote at \(x = 2\) |
| \(a = 3\) | B1 | From horizontal asymptote at \(y = 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f(x) > 3\) or \(y > 3\) | B1 | Either notation acceptable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt to rearrange: swap \(x\) and \(y\), multiply through by \((y-2)\) and collect terms | M1 | Look for minimum of cross multiplying by \((x-2)\) and proceeding to form a rearranged equation |
| \(f^{-1}(x) = \dfrac{2x+4}{x-3}\) | A1 | Accept \(y = \dfrac{2x+4}{x-3}\) |
| Domain: \(x > 3\) | A1 | Must be stated |
## Question 1:
### Part (a): State the values of $a$ and $b$
| Answer | Mark | Guidance |
|--------|------|----------|
| $b = 2$ | B1 | From vertical asymptote at $x = 2$ |
| $a = 3$ | B1 | From horizontal asymptote at $y = 3$ |
**(2 marks)**
### Part (b): State the range of f
| Answer | Mark | Guidance |
|--------|------|----------|
| $f(x) > 3$ or $y > 3$ | B1 | Either notation acceptable |
**(1 mark)**
### Part (c): Find $f^{-1}(x)$, stating its domain
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to rearrange: swap $x$ and $y$, multiply through by $(y-2)$ and collect terms | M1 | Look for minimum of cross multiplying by $(x-2)$ and proceeding to form a rearranged equation |
| $f^{-1}(x) = \dfrac{2x+4}{x-3}$ | A1 | Accept $y = \dfrac{2x+4}{x-3}$ |
| Domain: $x > 3$ | A1 | Must be stated |
**(3 marks)**
---\begin{enumerate}
\item The figure 1 shows part of the graph of $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { a x + 4 } { x - b } , \quad x > 2$
\end{enumerate}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f9dcb521-6aaa-4496-86e8-2dcd07838e10-02_837_1189_422_518}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
a. State the values of $a$ and $b$.\\
b. State the range of f.\\
c. Find $\mathrm { f } ^ { - 1 } ( x )$, stating its domain.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q1 [6]}}