| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Sketch exponential graphs |
| Difficulty | Standard +0.3 This is a multi-part question testing standard transformations of graphs, finding intercepts of exponential functions, and function composition. Parts (a)-(b) require routine sketching of inverse and transformations with labeled intercepts. Parts (c)-(d) involve straightforward substitution and solving 3(2^{-x}) - 1 = 0. Part (e) requires composing f and g, which simplifies nicely due to the inverse relationship between exponential and logarithm. All techniques are standard C3 material with no novel problem-solving required, though the multiple parts and careful attention to detail make it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06c Logarithm definition: log_a(x) as inverse of a^x |
\includegraphics{figure_3}
Figure 3 shows a sketch of the curve with equation $y = f(x), x \geq 0$. The curve meets the coordinate axes at the points $(0, c)$ and $(d, 0)$.
In separate diagrams sketch the curve with equation
\begin{enumerate}[label=(\alph*)]
\item $y = f^{-1}(x)$, [2]
\item $y = 3f(2x)$. [3]
\end{enumerate}
Indicate clearly on each sketch the coordinates, in terms of $c$ or $d$, of any point where the curve meets the coordinate axes.
Given that $f$ is defined by
$$f : x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item state
\begin{enumerate}[label=(\roman*)]
\item the value of $c$,
\item the range of $f$. [3]
\end{enumerate}
\item Find the value of $d$, giving your answer to 3 decimal places. [3]
\end{enumerate}
The function $g$ is defined by
$$g : x \to \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Find $fg(x)$, giving your answer in its simplest form. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q24 [14]}}