| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Moderate -0.3 This is a standard C3 composite and inverse functions question with routine procedures: finding range (straightforward from domain), finding inverse (algebraic manipulation of a square root function), and solving a composite equation. All parts follow textbook methods 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 |
2 The functions $f$ and $g$ are defined with their respective domains by
$$\begin{array} { l l }
f ( x ) = \sqrt { 2 x + 5 } , & \text { for real values of } x , x \geqslant - 2.5 \\
g ( x ) = \frac { 1 } { 4 x + 1 } , & \text { for real values of } x , x \neq - 0.25
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Find the range of f.
\item The inverse of f is $\mathrm { f } ^ { - 1 }$.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { f } ^ { - 1 } ( x )$.
\item State the domain of $\mathrm { f } ^ { - 1 }$.
\end{enumerate}\item The composite function fg is denoted by h .
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $\mathrm { h } ( x )$.
\item Solve the equation $\mathrm { h } ( x ) = 3$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2009 Q2 [10]}}