| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Standard +0.8 This question requires working backwards from a composite function fg(x) to find g(x), then computing gg(x). Students must recognize that fg(x) = f(g(x)) and deduce g(x) = 1/x by pattern matching, then apply this twice. This reverse-engineering approach and double composition goes beyond routine composite function exercises. |
| 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 | Mark | Guidance |
| \([f'(x)] = 16 - 2e^{2x}\) | B1 | |
| \(16 - 2e^{2x} = 0\) | M1 | For equating their derivative to zero (must be of form \(a + be^{2x}\)) |
| \(x = \dfrac{1}{2}\ln 8\) | A1 | Allow AWRT 1.04 |
| \([f(x) =]\ 8\ln 8 - 8\) | m1 | Correct subst of their \(x\) into \(f(x)\); Allow AWRT 8.63 or 8.64 |
| \(f(x) \leq 8\ln 8 - 8\) | A1 | Must have exact form and correct notation, no ISW; Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(g(x) = \dfrac{1}{x}\) | M1 | |
| \(gg(x) = x\) | A1 | NMS 2/2; Total: 2 |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[f'(x)] = 16 - 2e^{2x}$ | B1 | |
| $16 - 2e^{2x} = 0$ | M1 | For equating their derivative to zero (must be of form $a + be^{2x}$) |
| $x = \dfrac{1}{2}\ln 8$ | A1 | Allow AWRT 1.04 |
| $[f(x) =]\ 8\ln 8 - 8$ | m1 | Correct subst of their $x$ into $f(x)$; Allow AWRT 8.63 or 8.64 |
| $f(x) \leq 8\ln 8 - 8$ | A1 | Must have exact form and correct notation, no **ISW**; **Total: 5** |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $g(x) = \dfrac{1}{x}$ | M1 | |
| $gg(x) = x$ | A1 | **NMS** 2/2; **Total: 2** |
---5 The function f is defined by
$$\mathrm { f } ( x ) = 16 x - \mathrm { e } ^ { 2 x } , \text { for all real } x$$
The graph of $y = \mathrm { f } ( x )$ is sketched below.\\
\includegraphics[max width=\textwidth, alt={}, center]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-12_789_1349_534_347}
\begin{enumerate}[label=(\alph*)]
\item Find the range of f.
\item The composite function fg is defined by
$$\operatorname { fg } ( x ) = \frac { 16 } { x } - \mathrm { e } ^ { \frac { 2 } { x } } , \text { for real } x , x \neq 0$$
Find an expression for $\operatorname { gg } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2016 Q5 [7]}}