| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a slightly-above-average routine question on rational functions. Part (a) requires standard differentiation (quotient rule) and sign analysis. Part (b) is a textbook inverse function calculation. Part (c) involves composition of rational functions and determining range from the domain restriction—all standard techniques with no novel insight required. |
| 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 existence1.07h Differentiation from first principles: for sin(x) and cos(x)1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(x)=\frac{5x-3}{x-4}\Rightarrow f'(x)=\frac{5(x-4)-(5x-3)}{(x-4)^2}=\frac{k}{(x-4)^2}\) | M1 dM1 | M1: Quotient rule to form \(\frac{p(x-4)-q(5x-3)}{(x-4)^2}\); dM1: proceeds to \(f'(x)=\frac{k}{(x-4)^2}\) |
| \(f'(x)=\frac{-17}{(x-4)^2}<0\), hence decreasing | A1* | Requires correct \(f'(x)\), correct statement \(f'(x)<0\), and minimal conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y=\frac{5x-3}{x-4}\Rightarrow xy-4y=5x-3\Rightarrow x=\frac{4y-3}{y-5}\) | M1 | Cross multiply, collect \(x\) terms on one side |
| \(f^{-1}(x)=\frac{4x-3}{x-5}\) | A1 | Condone lhs as \(f^{-1}\) or \(y\) |
| Domain \(x>5\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(ff(x)=\dfrac{5\cdot\frac{5x-3}{x-4}-3}{\frac{5x-3}{x-4}-4}\) | M1 | Substitutes \(\frac{5x-3}{x-4}\) into \(f\); form must be correct |
| \(ff(x)=\frac{5(5x-3)-3(x-4)}{5x-3-4(x-4)}=\frac{22x-3}{x+13}\) | dM1 A1 | dM1: Multiplies numerator and denominator by \((x-4)\); A1: \(\frac{22x-3}{x+13}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
\(5| B1 B1 |
First B1: achieves one correct "end" value (not just the number \(5\)); Second B1: fully correct; accept \((5,22)\); condone non-strict inequalities \(5\leq ff<21\) for B1 B0 |
|
## Question 6:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(x)=\frac{5x-3}{x-4}\Rightarrow f'(x)=\frac{5(x-4)-(5x-3)}{(x-4)^2}=\frac{k}{(x-4)^2}$ | M1 dM1 | M1: Quotient rule to form $\frac{p(x-4)-q(5x-3)}{(x-4)^2}$; dM1: proceeds to $f'(x)=\frac{k}{(x-4)^2}$ |
| $f'(x)=\frac{-17}{(x-4)^2}<0$, hence decreasing | A1* | Requires correct $f'(x)$, correct statement $f'(x)<0$, and minimal conclusion |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y=\frac{5x-3}{x-4}\Rightarrow xy-4y=5x-3\Rightarrow x=\frac{4y-3}{y-5}$ | M1 | Cross multiply, collect $x$ terms on one side |
| $f^{-1}(x)=\frac{4x-3}{x-5}$ | A1 | Condone lhs as $f^{-1}$ or $y$ |
| Domain $x>5$ | B1 | |
### Part (c)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ff(x)=\dfrac{5\cdot\frac{5x-3}{x-4}-3}{\frac{5x-3}{x-4}-4}$ | M1 | Substitutes $\frac{5x-3}{x-4}$ into $f$; form must be correct |
| $ff(x)=\frac{5(5x-3)-3(x-4)}{5x-3-4(x-4)}=\frac{22x-3}{x+13}$ | dM1 A1 | dM1: Multiplies numerator and denominator by $(x-4)$; A1: $\frac{22x-3}{x+13}$ |
### Part (c)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5<ff(x)<22$ | B1 B1 | First B1: achieves one correct "end" value (not just the number $5$); Second B1: fully correct; accept $(5,22)$; condone non-strict inequalities $5\leq ff<21$ for B1 B0 |6. The function f is defined by
$$f ( x ) = \frac { 5 x - 3 } { x - 4 } \quad x > 4$$
\begin{enumerate}[label=(\alph*)]
\item Show, by using calculus, that f is a decreasing function.
\item Find $\mathrm { f } ^ { - 1 }$
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\mathrm { ff } ( x ) = \frac { a x + b } { x + c }$ where $a , b$ and $c$ are constants to be found.
\item Deduce the range of ff.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q6 [11]}}