| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 This is a straightforward composite functions question requiring standard techniques: identifying range from a polynomial, explaining why a many-to-one function has no inverse, forming a composite function by substitution, and solving a resulting equation. All parts are routine C3 material with no novel problem-solving 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 |
| \(f(x) \leq 2,\ f \leq 2,\ y \leq 2\) | B2 | \(\leq 2, f(x) < 2, x \leq 2\); \(y < 2,\ f < 2\) scores B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x)\) is not one to one | E1 | Allow "many to one" or numerical example |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(fg(x) = 2 - \left(\frac{1}{x-4}\right)^4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2 - \left(\frac{1}{x-4}\right)^4 = -14 \Rightarrow 16 = \left(\frac{1}{x-4}\right)^4\) | ||
| \((x-4)^4 = \frac{1}{16}\) | M1 | Correct handling of fourth root; must have \(\pm\) |
| \(x - 4 = \pm\frac{1}{2}\) | M1 | Correct handling of reciprocal |
| \(x = 4\tfrac{1}{2},\ 3\tfrac{1}{2}\) | A1 |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) \leq 2,\ f \leq 2,\ y \leq 2$ | B2 | $\leq 2, f(x) < 2, x \leq 2$; $y < 2,\ f < 2$ scores B1 |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x)$ is not one to one | E1 | Allow "many to one" or numerical example |
### Part (c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $fg(x) = 2 - \left(\frac{1}{x-4}\right)^4$ | B1 | |
### Part (c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2 - \left(\frac{1}{x-4}\right)^4 = -14 \Rightarrow 16 = \left(\frac{1}{x-4}\right)^4$ | | |
| $(x-4)^4 = \frac{1}{16}$ | M1 | Correct handling of fourth root; must have $\pm$ |
| $x - 4 = \pm\frac{1}{2}$ | M1 | Correct handling of reciprocal |
| $x = 4\tfrac{1}{2},\ 3\tfrac{1}{2}$ | A1 | |
**Total: 7 marks**5 The functions $f$ and $g$ are defined with their respective domains by
$$\begin{array} { l l }
\mathrm { f } ( x ) = 2 - x ^ { 4 } & \text { for all real values of } x \\
\mathrm {~g} ( x ) = \frac { 1 } { x - 4 } & \text { for real values of } x , x \neq 4
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item State the range of f .
\item Explain why the function f does not have an inverse.
\item \begin{enumerate}[label=(\roman*)]
\item Write down an expression for fg(x).
\item Solve the equation $\operatorname { fg } ( x ) = - 14$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2009 Q5 [7]}}