| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Moderate -0.8 This is a straightforward composite and inverse functions question requiring only routine techniques: stating range of a cubic, computing fg(x) by substitution, solving a simple equation, finding inverse by swapping and rearranging, and stating its range. All parts are standard textbook exercises with no problem-solving or novel insight required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| all (real) values | B1 | No \(x\) in answer, unless \(f(x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(f_g(x) = \left(\frac{1}{x-3}\right)^3\) | B1 | ISW |
| \(\left(\frac{1}{x-3}\right)^3 = 64\) | M1 | \(\sqrt[3]{...}\) |
| \(\frac{1}{x-3} = 4\) | ||
| \(x - 3 = \frac{1}{4}\) | M1 | Invert |
| \(x = 3\frac{1}{4}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \frac{1}{x-3}\) | M1 | Swap \(x\) and \(y\) |
| \(x = \frac{1}{y-3}\) | ||
| \(x(y-3) = 1\) | M1 | attempt to isolate |
| \(xy - 3x = 1\) | ||
| \(y = \frac{1 + 3x}{x} = g^{-1}(x)\) or \(\frac{1}{x} + 3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| (real values) \((g^{-1}(x)) \neq 3\) | B1 |
### 4(a)
all (real) values | B1 | No $x$ in answer, unless $f(x)$ | **Total: 1**
### 4(b)(i)
$f_g(x) = \left(\frac{1}{x-3}\right)^3$ | B1 | ISW |
$\left(\frac{1}{x-3}\right)^3 = 64$ | M1 | $\sqrt[3]{...}$ |
$\frac{1}{x-3} = 4$ | | |
$x - 3 = \frac{1}{4}$ | M1 | Invert |
$x = 3\frac{1}{4}$ | A1 | | **Total: 3**
### 4(c)(i)
$y = \frac{1}{x-3}$ | M1 | Swap $x$ and $y$ |
$x = \frac{1}{y-3}$ | | |
$x(y-3) = 1$ | M1 | attempt to isolate |
$xy - 3x = 1$ | | |
$y = \frac{1 + 3x}{x} = g^{-1}(x)$ or $\frac{1}{x} + 3$ | A1 | | **Total: 3**
### 4(c)(ii)
(real values) $(g^{-1}(x)) \neq 3$ | B1 | | **Total: 1**4 The functions f and g are defined with their respective domains by
$$\begin{array} { l l }
\mathrm { f } ( x ) = x ^ { 3 } , & \text { for all real values of } x \\
\mathrm {~g} ( x ) = \frac { 1 } { x - 3 } , & \text { for real values of } x , x \neq 3
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item State the range of f.
\item \begin{enumerate}[label=(\roman*)]
\item Find fg(x).
\item Solve the equation $\operatorname { fg } ( x ) = 64$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item The inverse of g is $\mathrm { g } ^ { - 1 }$. Find $\mathrm { g } ^ { - 1 } ( x )$.
\item State the range of $\mathrm { g } ^ { - 1 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2008 Q4 [9]}}