| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Verify composite identity |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 topics (inverse functions, composite functions, transformations). Part (a) requires algebraic manipulation to find the inverse and verify equality—a routine technique. Part (b) uses the self-inverse property discovered in (a). Parts (c)-(e) involve reading values from a graph and applying standard transformations (reflection for inverse, horizontal shift and vertical stretch for h). While multi-step, each component is a textbook exercise requiring no novel insight, making it slightly easier than the average A-level question. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
The function $f$ is defined by $f: x \mapsto \frac{3x-1}{x-3}, x \in \mathbb{R}, x \neq 3$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $f^{-1}(x) = f(x)$ for all $x \in \mathbb{R}, x \neq 3$. [3]
\item Hence find, in terms of $k$, $ff(k)$, where $x \neq 3$. [2]
\end{enumerate}
\includegraphics{figure_3}
Figure 3 shows a sketch of the one-one function $g$, defined over the domain $-2 \leq x \leq 2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the value of $fg(-2)$. [3]
\item Sketch the graph of the inverse function $g^{-1}$ and state its domain. [3]
\end{enumerate}
The function $h$ is defined by $h: x \mapsto 2g(x - 1)$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Sketch the graph of the function $h$ and state its range. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q19 [14]}}