| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Verify composite identity |
| Difficulty | Standard +0.3 This is a standard C3 functions question covering inverse functions, function composition, and transformations. Part (a) requires routine algebraic manipulation to find the inverse. Parts (b)-(e) involve straightforward function composition and graph transformations that are typical textbook exercises. The self-inverse property is elegant but not conceptually demanding, and all techniques are standard for this module. |
| Spec | 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 $\text{f}: x \to \frac{3x - 1}{x - 3}$, $x \in \mathbb{R}$, $x \neq 3$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $\text{f}^{-1}(x) = \text{f}(x)$ for all $x \in \mathbb{R}$, $x \neq 3$. [3]
\item Hence find, in terms of $k$, $\text{f}f(k)$, 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 $\text{f}g(-2)$. [3]
\item Sketch the graph of the inverse function $\text{g}^{-1}$ and state its domain. [3]
\end{enumerate}
The function h is defined by $\text{h}: x \mapsto 2g(x - 1)$.
\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 Q6 [14]}}