| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a standard C3 inverse function question with routine techniques: finding inverse by swapping x and y, determining domain/range, and composing functions. Part (a) requires basic logarithm manipulation, (b) is straightforward range-to-domain conversion, (c) involves direct substitution that simplifies nicely, and (d) requires recognizing that e^(x²) ≥ 1. All parts use well-practiced methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 4 - \ln(x+2)\), \(\ln(x+2) = 4-y\), \(x+2 = e^{4-y}\) | M1 | |
| \(x = e^{4-y} - 2\), so \(f^{-1}(x) = e^{4-x} - 2\) | M1A1 | oe |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x \leq 4\) | B1 | |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(fg(x) = 4 - \ln(e^{x^2} - 2 + 2)\) | M1 | |
| \(fg(x) = 4 - x^2\) | dM1A1 | |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(fg(x) \leq 4\) | B1ft | |
| (1) | 8 Marks |
# Question 4:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 4 - \ln(x+2)$, $\ln(x+2) = 4-y$, $x+2 = e^{4-y}$ | M1 | |
| $x = e^{4-y} - 2$, so $f^{-1}(x) = e^{4-x} - 2$ | M1A1 | oe |
| | (3) | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x \leq 4$ | B1 | |
| | (1) | |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $fg(x) = 4 - \ln(e^{x^2} - 2 + 2)$ | M1 | |
| $fg(x) = 4 - x^2$ | dM1A1 | |
| | (3) | |
## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $fg(x) \leq 4$ | B1ft | |
| | (1) | **8 Marks** |
---4. The function $f$ is defined by
$$\mathrm { f } : x \mapsto 4 - \ln ( x + 2 ) , \quad x \in \mathbb { R } , x \geqslant - 1$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { f } ^ { - 1 } ( x )$.
\item Find the domain of $\mathrm { f } ^ { - 1 }$.
The function $g$ is defined by
$$\mathrm { g } : x \mapsto \mathrm { e } ^ { x ^ { 2 } } - 2 , \quad x \in \mathbb { R }$$
\item Find $\mathrm { fg } ( x )$, giving your answer in its simplest form.
\item Find the range of fg.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2011 Q4 [8]}}