| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2020 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine domain for composite |
| Difficulty | Challenging +1.8 This AEA question requires understanding of composite function domains, exponential/logarithmic manipulation to simplify gf(x), finding where a circle is tangent to a line (perpendicular distance formula), and multi-step reasoning across parts. While systematic, it demands careful domain analysis and geometric insight beyond standard A-level, though the algebraic manipulations themselves are routine once the approach is identified. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties |
2.The functions $f$ and $g$ are defined by
$$\begin{array} { l l }
\mathrm { f } ( x ) = 2 \sqrt { 1 - \mathrm { e } ^ { - x } } & x \in \mathbb { R } , x \geqslant 0 \\
\mathrm {~g} ( x ) = \ln \left( 4 - x ^ { 2 } \right) & x \in \mathbb { R } , - 2 < x < 2
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Explain why fg cannot be formed as a composite function.
\item Explain why gf can be formed as a composite function.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { gf } ( x )$ ,giving the answer in the form $\mathrm { gf } ( x ) = a + b x$ ,where $a$ and $b$ are constants.
\item State the domain and range of gf.
\end{enumerate}\item Sketch the graph of the function gf.\\
On your sketch,you should show the coordinates of any points where the graph meets or crosses the coordinate axes.
The circle $C$ with centre $( 0 , - \ln 9 )$ touches the line with equation $y = \operatorname { gf } ( x )$ at precisely one point.
\item Find an equation of the circle $C$ .
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2020 Q2 [13]}}