| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Moderate -0.3 This is a straightforward C3 functions question testing standard concepts: explaining why a function lacks an inverse (many-to-one), finding an inverse by swapping and rearranging, stating range from domain, and solving a composite function equation. All parts are routine textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02v Inverse and composite functions: graphs and conditions for existence |
The functions f and g are defined with their respective domains by
$$f(x) = x^2 \quad \text{for all real values of } x$$
$$g(x) = \frac{1}{2x + 1} \quad \text{for real values of } x, \quad x \neq -0.5$$
\begin{enumerate}[label=(\alph*)]
\item Explain why f does not have an inverse. [1]
\item The inverse of g is $g^{-1}$. Find $g^{-1}(x)$. [3]
\item State the range of $g^{-1}$. [1]
\item Solve the equation $fg(x) = g(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2011 Q5 [8]}}