| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 Part (a) is a standard inverse function calculation for a rational function requiring algebraic manipulation. Parts (b) and (c) are routine verification and range determination. This is a typical textbook exercise slightly easier than average due to its straightforward algebraic nature and clear structure. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \frac{2x}{3x-1} \rightarrow 3xy - y = 2x \rightarrow 3xy - 2x = y\) (or \(-y = 2x - 3xy\)) | *M1 | For 1st two operations. Condone a sign error |
| \(x(3y-2) = y \rightarrow x = \frac{y}{3y-2}\) (or \(x = \frac{-y}{2-3y}\)) | DM1 | For 2nd two operations. Condone a sign error |
| \(f^{-1}(x) = \frac{x}{3x-2}\) | A1 | Allow \(f^{-1}(x) = \frac{-x}{2-3x}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[\frac{2(3x-1)+2}{3(3x-1)}\right] = \left[\frac{6x}{3(3x-1)}\right] = \left[\frac{2x}{3x-1}\right]\) | B1 B1 | AG, WWW. First B1 for a correct single unsimplified fraction. An intermediate step needs to be shown. Equivalent methods accepted |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) > \frac{2}{3}\) | B1 | Allow \((y) > \frac{2}{3}\). Do not allow \(x > \frac{2}{3}\) |
## Question 6(a):
| $y = \frac{2x}{3x-1} \rightarrow 3xy - y = 2x \rightarrow 3xy - 2x = y$ (or $-y = 2x - 3xy$) | *M1 | For 1st two operations. Condone a sign error |
|---|---|---|
| $x(3y-2) = y \rightarrow x = \frac{y}{3y-2}$ (or $x = \frac{-y}{2-3y}$) | DM1 | For 2nd two operations. Condone a sign error |
| $f^{-1}(x) = \frac{x}{3x-2}$ | A1 | Allow $f^{-1}(x) = \frac{-x}{2-3x}$ |
## Question 6(b):
| $\left[\frac{2(3x-1)+2}{3(3x-1)}\right] = \left[\frac{6x}{3(3x-1)}\right] = \left[\frac{2x}{3x-1}\right]$ | B1 B1 | AG, WWW. First B1 for a correct single unsimplified fraction. An intermediate step needs to be shown. Equivalent methods accepted |
|---|---|---|
## Question 6(c):
| $f(x) > \frac{2}{3}$ | B1 | Allow $(y) > \frac{2}{3}$. Do not allow $x > \frac{2}{3}$ |
|---|---|---|
---6 The function f is defined by $\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 1 }$ for $x > \frac { 1 } { 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\item Show that $\frac { 2 } { 3 } + \frac { 2 } { 3 ( 3 x - 1 ) }$ can be expressed as $\frac { 2 x } { 3 x - 1 }$.
\item State the range of f.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q6 [6]}}