| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a standard C3/C4 inverse function question with routine algebraic manipulation. Part (a) requires finding range using limits or rearrangement, part (b) is the standard swap-and-solve method for finding inverses, and part (c) involves function composition with rational functions. All techniques are textbook exercises with no novel insight required, making it slightly 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 | Marks | Guidance |
| \(0 < g < 3\) | M1A1 | M1 for one correct end; accept \(0 < g < 3\), \(0 < y < 3\), \(g(x)>0\) and \(g(x)<3\), \((0,3)\). Not \(x>0\) or \(x<3\) on their own |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = \frac{6x}{2x+3} \Rightarrow 2xy+3y=6x \Rightarrow (6-2y)x = 3y \Rightarrow x = \frac{3y}{(6-2y)}\) | M1A1 | Attempt to make \(x\) subject; cross multiply, expand, collect/factorise |
| \(g^{-1}(x) = \frac{3x}{(6-2x)}, \quad 0 < x < 3\) | A1ft | Also accept \(\frac{-3x}{(2x-6)}\); domain in terms of \(x\); follow through on range from (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(gg(x) = g\!\left(\frac{6x}{2x+3}\right) = \dfrac{6 \times \dfrac{6x}{2x+3}}{2\times\dfrac{6x}{2x+3}+3}\) | M1 | Attempts \(g\!\left(\frac{6x}{2x+3}\right)\) |
| \(= \dfrac{6 \times 6x}{2\times 6x + 3(2x+3)}\) | dM1 | Correct processing to single fraction \(\frac{a}{b}\) |
| \(= \dfrac{36x}{18x+9} = \dfrac{4x}{2x+1}\) | A1 | cao; ignore presence/absence of domain |
## Question 3:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0 < g < 3$ | M1A1 | M1 for one correct end; accept $0 < g < 3$, $0 < y < 3$, $g(x)>0$ and $g(x)<3$, $(0,3)$. Not $x>0$ or $x<3$ on their own |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = \frac{6x}{2x+3} \Rightarrow 2xy+3y=6x \Rightarrow (6-2y)x = 3y \Rightarrow x = \frac{3y}{(6-2y)}$ | M1A1 | Attempt to make $x$ subject; cross multiply, expand, collect/factorise |
| $g^{-1}(x) = \frac{3x}{(6-2x)}, \quad 0 < x < 3$ | A1ft | Also accept $\frac{-3x}{(2x-6)}$; domain in terms of $x$; follow through on range from (a) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $gg(x) = g\!\left(\frac{6x}{2x+3}\right) = \dfrac{6 \times \dfrac{6x}{2x+3}}{2\times\dfrac{6x}{2x+3}+3}$ | M1 | Attempts $g\!\left(\frac{6x}{2x+3}\right)$ |
| $= \dfrac{6 \times 6x}{2\times 6x + 3(2x+3)}$ | dM1 | Correct processing to single fraction $\frac{a}{b}$ |
| $= \dfrac{36x}{18x+9} = \dfrac{4x}{2x+1}$ | A1 | cao; ignore presence/absence of domain |3. The function g is defined by
$$g ( x ) = \frac { 6 x } { 2 x + 3 } \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find the range of g .
\item Find $\mathrm { g } ^ { - 1 } ( x )$ and state its domain.
\item Find the function $\operatorname { gg } ( x )$, writing your answer as a single fraction in its simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2017 Q3 [8]}}