| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2020 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Verify composite identity |
| Difficulty | Moderate -0.3 Part (a) requires finding the inverse function and evaluating it at a point, which is a standard technique. Part (b) involves composing a rational function with itself and simplifying—straightforward algebraic manipulation with no novel insight required. This is slightly easier than average due to being routine practice of core techniques. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
| VI4V SIHI NI ILIUM ION OC | VIAV SIHI NI III IM I O N OO | VJAV SIHI NI III M M ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either attempts \(\frac{3x-7}{x-2}=7 \Rightarrow x=\ldots\) or attempts \(f^{-1}(x)\) and substitutes \(x=7\) | M1 | Look for attempt to multiply by \((x-2)\) leading to value for \(x\). Method for finding \(f^{-1}(x)\) should be sound, condone slips. FYI \(f^{-1}(x)=\frac{2x-7}{x-3}\) |
| \(\frac{7}{4}\) oe | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(ff(x) = \dfrac{3\times\left(\frac{3x-7}{x-2}\right)-7}{\left(\frac{3x-7}{x-2}\right)-2} = \dfrac{3(3x-7)-7(x-2)}{3x-7-2(x-2)}\) | M1 | Attempt at fully substituting \(\frac{3x-7}{x-2}\) into \(f(x)\). Expression must have correct form. Condone slips. |
| dM1 | Attempts to multiply all terms in numerator and denominator by \((x-2)\) to create fraction \(\frac{P(x)}{Q(x)}\) where both \(P(x)\) and \(Q(x)\) are linear. Condone \(\frac{P(x)}{Q(x)}\times\frac{x-2}{x-2}\) | |
| \(= \dfrac{2x-7}{x-3}\) | A1 | Reached via careful and accurate work. Implied by \(a=2, b=-7\) following correct work. |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Either attempts $\frac{3x-7}{x-2}=7 \Rightarrow x=\ldots$ or attempts $f^{-1}(x)$ and substitutes $x=7$ | M1 | Look for attempt to multiply by $(x-2)$ leading to value for $x$. Method for finding $f^{-1}(x)$ should be sound, condone slips. FYI $f^{-1}(x)=\frac{2x-7}{x-3}$ |
| $\frac{7}{4}$ oe | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $ff(x) = \dfrac{3\times\left(\frac{3x-7}{x-2}\right)-7}{\left(\frac{3x-7}{x-2}\right)-2} = \dfrac{3(3x-7)-7(x-2)}{3x-7-2(x-2)}$ | M1 | Attempt at fully substituting $\frac{3x-7}{x-2}$ into $f(x)$. Expression must have correct form. Condone slips. |
| | dM1 | Attempts to multiply all terms in numerator and denominator by $(x-2)$ to create fraction $\frac{P(x)}{Q(x)}$ where both $P(x)$ and $Q(x)$ are linear. Condone $\frac{P(x)}{Q(x)}\times\frac{x-2}{x-2}$ |
| $= \dfrac{2x-7}{x-3}$ | A1 | Reached via careful and accurate work. Implied by $a=2, b=-7$ following correct work. |
---\begin{enumerate}
\item The function f is defined by
\end{enumerate}
$$f ( x ) = \frac { 3 x - 7 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$
(a) Find $f ^ { - 1 } ( 7 )$\\
(b) Show that $\operatorname { ff } ( x ) = \frac { a x + b } { x - 3 }$ where $a$ and $b$ are integers to be found.
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VI4V SIHI NI ILIUM ION OC & VIAV SIHI NI III IM I O N OO & VJAV SIHI NI III M M ION OC \\
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\hfill \mbox{\textit{Edexcel Paper 1 2020 Q4 [5]}}