| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a structured multi-part question with clear guidance. Part (a) involves algebraic manipulation (factorizing the quadratic denominator and combining fractions) which is routine but requires care. Part (b) is a standard inverse function procedure once the simplified form is obtained. Part (c) requires understanding that the domain of f^{-1} equals the range of f, which is straightforward given x > 4. The question is slightly above average due to the algebraic complexity in part (a), but the scaffolding and standard techniques make it accessible. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(2x^2-3x-5=(2x-5)(x+1)\) | B1 | Correct factorisation, can be scored anywhere. Condone \(2(x-2.5)(x+1)\) |
| \(3-\frac{x-2}{x+1}+\frac{5x+26}{2x^2-3x-5} = \frac{3(x+1)(2x-5)-(x-2)(2x-5)+5x+26}{(x+1)(2x-5)}\) | M1 A1 | M1: Attempt to combine all three terms using common denominator; at least one numerator adapted correctly. A1: Correct fraction with denominator \((x+1)(2x-5)\) or equivalent |
| \(= \frac{(4x+1)(x+1)}{(x+1)(2x-5)} = \frac{4x+1}{2x-5}\) | A1 | Correct simplified fraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Correct attempt at inverse: \(y=\frac{4x+1}{2x-5} \Rightarrow x = \ldots\) | M1 | Attempt to change subject; look for cross-multiplying by \(cx+d\) and proceeding to \(x=g(y)\). Allow if one of \(a,b\) or \(d=0\) |
| \(f^{-1}(x) = \frac{5x+1}{2x-4}\) | A1 | Condone \(y=\frac{5x+1}{2x-4}\) and \(f^{-1}=\frac{5x+1}{2x-4}\). Allow equivalents e.g. \(\frac{-5x-1}{4-2x}\), \(\frac{-\frac{5}{2}x-\frac{1}{2}}{2-x}\), \(\frac{5}{2}+\frac{11}{2x-4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(2 < x < \frac{17}{3}\) | M1 A1 | M1: Finding one "end" of the domain (must be numerical). Sight of \(\frac{17}{3}\) or \(f(4)\), or 2 or \(f(x)\) as \(x\to\infty\). A1: Correct domain with allowable notation e.g. \(\left(2,\frac{17}{3}\right)\), \(x>2\) and \(x<\frac{17}{3}\). Condone "or" |
# Question 3(a):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $2x^2-3x-5=(2x-5)(x+1)$ | B1 | Correct factorisation, can be scored anywhere. Condone $2(x-2.5)(x+1)$ |
| $3-\frac{x-2}{x+1}+\frac{5x+26}{2x^2-3x-5} = \frac{3(x+1)(2x-5)-(x-2)(2x-5)+5x+26}{(x+1)(2x-5)}$ | M1 A1 | M1: Attempt to combine all three terms using common denominator; at least one numerator adapted correctly. A1: Correct fraction with denominator $(x+1)(2x-5)$ or equivalent |
| $= \frac{(4x+1)(x+1)}{(x+1)(2x-5)} = \frac{4x+1}{2x-5}$ | A1 | Correct simplified fraction |
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# Question 3(b):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Correct attempt at inverse: $y=\frac{4x+1}{2x-5} \Rightarrow x = \ldots$ | M1 | Attempt to change subject; look for cross-multiplying by $cx+d$ and proceeding to $x=g(y)$. Allow if one of $a,b$ or $d=0$ |
| $f^{-1}(x) = \frac{5x+1}{2x-4}$ | A1 | Condone $y=\frac{5x+1}{2x-4}$ and $f^{-1}=\frac{5x+1}{2x-4}$. Allow equivalents e.g. $\frac{-5x-1}{4-2x}$, $\frac{-\frac{5}{2}x-\frac{1}{2}}{2-x}$, $\frac{5}{2}+\frac{11}{2x-4}$ |
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# Question 3(c):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $2 < x < \frac{17}{3}$ | M1 A1 | M1: Finding one "end" of the domain (must be numerical). Sight of $\frac{17}{3}$ or $f(4)$, or 2 or $f(x)$ as $x\to\infty$. A1: Correct domain with allowable notation e.g. $\left(2,\frac{17}{3}\right)$, $x>2$ and $x<\frac{17}{3}$. Condone "or" |
---3.
$$f ( x ) = 3 - \frac { x - 2 } { x + 1 } + \frac { 5 x + 26 } { 2 x ^ { 2 } - 3 x - 5 } \quad x > 4$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\mathrm { f } ( x ) = \frac { a x + b } { c x + d } \quad x > 4$$
where $a , b , c$ and $d$ are integers to be found.
\item Hence find $\mathrm { f } ^ { - 1 } ( x )$
\item Find the domain of $\mathrm { f } ^ { - 1 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2021 Q3 [8]}}