| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Moderate -0.8 Part (a) is straightforward function composition requiring simple substitution. Part (b) is routine inverse function finding for a rational function. Part (c) requires solving an equation involving both functions but follows standard algebraic manipulation. All parts are textbook-standard exercises with no 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/Working | Mark | Guidance |
| \(fg(5) = f(3) = \frac{5-\text{"3"}}{3\times\text{"3"}+2} = \frac{2}{11}\) | M1, A1 | Correct order: g before f on 5; allow two-step approach \(g(5)=A \rightarrow f(A)=\ldots\); condone arithmetic slips; A1 for \(\frac{2}{11}\) or exact equivalent, accept \(0.1\dot{8}\) but not \(0.18\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f^{-1}(x) = \frac{5-2x}{3x+1}\) | M1A1 | M1 for changing subject of \(y=\frac{5-x}{3x+2}\) to get \(x=\frac{\ldots\pm2y}{\pm3y\pm\ldots}\); A1 for \(\frac{5-2x}{3x+1}\) or equivalent e.g. \(\frac{2x-5}{-3x-1}\); accept mapping notation; do not allow just \(y=\ldots\) |
| \(x \in \mathbb{R},\ x \neq -\frac{1}{3}\) | B1 | Omission of \(x \in \mathbb{R}\) condoned |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{5-\frac{1}{a}}{3\times\frac{1}{a}+2} = 2(a+3)-7\) | B1ft | Follow through on their \(f^{-1}(x)\); or equivalent \(\frac{1}{a} = f^{-1}(2a-1)\) |
| \(\frac{5-\frac{1}{a}}{3\times\frac{1}{a}+2} = 2(a+3)-7 \Rightarrow 4a^2-a-2=0\) | M1A1 | Must start with allowable equation form; rearranges to 3-term quadratic; A1 for \(4a^2-a-2=0\) or equivalent |
| \(a = \frac{1\pm\sqrt{33}}{8}\) | A1 | Exact equivalent only; do not accept rounded decimals |
## Question 2:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $fg(5) = f(3) = \frac{5-\text{"3"}}{3\times\text{"3"}+2} = \frac{2}{11}$ | M1, A1 | Correct order: g before f on 5; allow two-step approach $g(5)=A \rightarrow f(A)=\ldots$; condone arithmetic slips; A1 for $\frac{2}{11}$ or exact equivalent, accept $0.1\dot{8}$ but not $0.18$ |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f^{-1}(x) = \frac{5-2x}{3x+1}$ | M1A1 | M1 for changing subject of $y=\frac{5-x}{3x+2}$ to get $x=\frac{\ldots\pm2y}{\pm3y\pm\ldots}$; A1 for $\frac{5-2x}{3x+1}$ or equivalent e.g. $\frac{2x-5}{-3x-1}$; accept mapping notation; do not allow just $y=\ldots$ |
| $x \in \mathbb{R},\ x \neq -\frac{1}{3}$ | B1 | Omission of $x \in \mathbb{R}$ condoned |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{5-\frac{1}{a}}{3\times\frac{1}{a}+2} = 2(a+3)-7$ | B1ft | Follow through on their $f^{-1}(x)$; or equivalent $\frac{1}{a} = f^{-1}(2a-1)$ |
| $\frac{5-\frac{1}{a}}{3\times\frac{1}{a}+2} = 2(a+3)-7 \Rightarrow 4a^2-a-2=0$ | M1A1 | Must start with allowable equation form; rearranges to 3-term quadratic; A1 for $4a^2-a-2=0$ or equivalent |
| $a = \frac{1\pm\sqrt{33}}{8}$ | A1 | Exact equivalent only; do not accept rounded decimals |
---2. The functions f and g are defined by
$$\begin{array} { l l }
\mathrm { f } ( x ) = \frac { 5 - x } { 3 x + 2 } & x \in \mathbb { R } , x \neq - \frac { 2 } { 3 } \\
\mathrm {~g} ( x ) = 2 x - 7 & x \in \mathbb { R }
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\mathrm { fg } ( 5 )$
\item Find $\mathrm { f } ^ { - 1 }$
\item Solve the equation
$$\mathrm { f } \left( \frac { 1 } { a } \right) = \mathrm { g } ( a + 3 )$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q2 [9]}}