| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Verify composite identity |
| Difficulty | Moderate -0.8 This is a straightforward composite function question requiring routine substitution and basic calculus. Part (a) involves simple algebraic manipulation with exponentials and logarithms, parts (b)-(c) are standard curve sketching, and part (d) is basic differentiation and equation solving. All techniques are standard C3 material with no novel problem-solving required. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06c Logarithm definition: log_a(x) as inverse of a^x1.06g Equations with exponentials: solve a^x = b |
8. The functions $f$ and $g$ are defined by
$$\begin{array} { l l }
\mathrm { f } : x \mapsto 2 x + \ln 2 , & x \in \mathbb { R } , \\
\mathrm {~g} : x \mapsto \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } .
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Prove that the composite function gf is
$$\operatorname { gf } : x \mapsto 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
\item Sketch the curve with equation $y = \operatorname { gf } ( x )$, and show the coordinates of the point where the curve cuts the $y$-axis.
\item Write down the range of gf .
\item Find the value of $x$ for which $\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3$, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q8 [13]}}