| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a straightforward P3 question testing standard inverse function techniques. Part (a) requires finding range by considering behavior as x→∞ and at x=0. Part (b) involves routine algebraic manipulation to find the inverse (rearranging a rational function). Part (c) is simple function composition with substitution. All parts are textbook exercises requiring no novel insight, making this slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either \(f(x)<5\) or \(f(x)\ldots 3\) | M1 | One correct end of range; allow \(\leq\) or \(<\) in correct direction |
| \(3 \leq f(x) < 5\) | A1 | Correct range; allow with \(y\) instead of \(f(x)\); accept two separate inequalities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y=5-\frac{4}{3x+2} \Rightarrow \frac{4}{3x+2}=5-y \Rightarrow \frac{4}{5-y}=3x+2 \Rightarrow x=...\) | M1 | Attempts to make \(x\) subject; allow sign slips but correct order of operations |
| \(f^{-1}(x) = \frac{1}{3}\!\left(\frac{4}{5-x}-2\right)\) or equivalent e.g. \(\frac{4}{15-3x}-\frac{2}{3}\) or \(\frac{2x-6}{15-3x}\) | A1 | Correct rule in terms of \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Domain is \(3 \leq x < 5\) | B1ft | Follow through on answer to (a); accept interval or set notation; do not accept \(f^{-1}(x)<5\) or with \(y\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{fg}(-\pi)=f\!\left(\left\lvert4\sin\!\left(-\frac{\pi}{3}+\frac{\pi}{6}\right)\right\rvert\right)=f\!\left(\left\lvert4\sin\!\left(-\frac{\pi}{6}\right)\right\rvert\right)=f(2)=...\) | M1 | Attempts to evaluate \(g\) at \(-\pi\) and substitutes into \(f\); evaluation of \(g(-\pi)\) must be attempted |
| \(=5-\frac{4}{6+2}=\frac{9}{2}\) | A1 | Correct answer; accept as decimal; do not isw if changed to degrees |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Either $f(x)<5$ or $f(x)\ldots 3$ | M1 | One correct end of range; allow $\leq$ or $<$ in correct direction |
| $3 \leq f(x) < 5$ | A1 | Correct range; allow with $y$ instead of $f(x)$; accept two separate inequalities |
## Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y=5-\frac{4}{3x+2} \Rightarrow \frac{4}{3x+2}=5-y \Rightarrow \frac{4}{5-y}=3x+2 \Rightarrow x=...$ | M1 | Attempts to make $x$ subject; allow sign slips but correct order of operations |
| $f^{-1}(x) = \frac{1}{3}\!\left(\frac{4}{5-x}-2\right)$ or equivalent e.g. $\frac{4}{15-3x}-\frac{2}{3}$ or $\frac{2x-6}{15-3x}$ | A1 | Correct rule in terms of $x$ |
## Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Domain is $3 \leq x < 5$ | B1ft | Follow through on answer to (a); accept interval or set notation; do not accept $f^{-1}(x)<5$ or with $y$ |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{fg}(-\pi)=f\!\left(\left\lvert4\sin\!\left(-\frac{\pi}{3}+\frac{\pi}{6}\right)\right\rvert\right)=f\!\left(\left\lvert4\sin\!\left(-\frac{\pi}{6}\right)\right\rvert\right)=f(2)=...$ | M1 | Attempts to evaluate $g$ at $-\pi$ and substitutes into $f$; evaluation of $g(-\pi)$ must be attempted |
| $=5-\frac{4}{6+2}=\frac{9}{2}$ | A1 | Correct answer; accept as decimal; do not isw if changed to degrees |
---2. The functions f and g are defined by
$$\begin{array} { l l }
f ( x ) = 5 - \frac { 4 } { 3 x + 2 } & x \geqslant 0 \\
g ( x ) = \left| 4 \sin \left( \frac { x } { 3 } + \frac { \pi } { 6 } \right) \right| & x \in \mathbb { R }
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Find the range of f
\item \begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { f } ^ { - 1 } ( x )$
\item Write down the domain of $\mathrm { f } ^ { - 1 }$
\end{enumerate}\item Find $\mathrm { fg } ( - \pi )$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q2 [7]}}