| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a straightforward C3 inverse function question requiring standard techniques: identifying range from exponential properties, finding inverse by swapping and rearranging (with the answer given to verify), and differentiating the inverse. All steps are routine applications of core methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = 2e^{3x} - 1\); Range: \(f(x) > -1\) (or \(y > -1\) or \(f > -1\)) | M1 A1 | for \(-1\) only; exactly correct; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 2e^{3x} - 1\); \(x = 2e^{3y} - 1\); \(2e^{3y} = x + 1\); \(e^{3y} = \frac{x+1}{2}\); \(y = \frac{1}{3}\ln\left(\frac{x+1}{2}\right)\) | M1 M1 M1 A1 | \(x \leftrightarrow y\); attempt to isolate; all correct with no error AG (be convinced); 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(f^{-1}(x) = \frac{1}{3}\left(\frac{2}{x+1}\right) \times \frac{1}{2}\) OE; for \(\frac{1}{2}\); all correct; \(x = 0\); \(f^{-1}(0) = \frac{1}{3}\) | M1 A1 A1 A1 | for differentiation of ln; their \((x+1)\); CSO; 4 marks total |
| Alternative: \(f^{-1}(x) = \frac{1}{3}\ln(x+1) - \frac{1}{3}\ln 2\); \(f^{-1'}(x) = \frac{1}{3(x+1)}\); \(f^{-1'}(0) = \frac{1}{3}\) | M1A1 A1 | CSO; 3 marks total |
### 8(a)
$f(x) = 2e^{3x} - 1$; Range: $f(x) > -1$ (or $y > -1$ or $f > -1$) | M1 A1 | for $-1$ only; exactly correct; 2 marks total
### 8(b)
$y = 2e^{3x} - 1$; $x = 2e^{3y} - 1$; $2e^{3y} = x + 1$; $e^{3y} = \frac{x+1}{2}$; $y = \frac{1}{3}\ln\left(\frac{x+1}{2}\right)$ | M1 M1 M1 A1 | $x \leftrightarrow y$; attempt to isolate; all correct with no error AG (be convinced); 3 marks total
### 8(c)
$f^{-1}(x) = \frac{1}{3}\left(\frac{2}{x+1}\right) \times \frac{1}{2}$ OE; for $\frac{1}{2}$; all correct; $x = 0$; $f^{-1}(0) = \frac{1}{3}$ | M1 A1 A1 A1 | for differentiation of ln; their $(x+1)$; CSO; 4 marks total
**Alternative:** $f^{-1}(x) = \frac{1}{3}\ln(x+1) - \frac{1}{3}\ln 2$; $f^{-1'}(x) = \frac{1}{3(x+1)}$; $f^{-1'}(0) = \frac{1}{3}$ | M1A1 A1 | CSO; 3 marks total
**Total for Question 8:** 9 marks8 A function f is defined by $\mathrm { f } ( x ) = 2 \mathrm { e } ^ { 3 x } - 1$ for all real values of $x$.
\begin{enumerate}[label=(\alph*)]
\item Find the range of f.
\item Show that $\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 } { 3 } \ln \left( \frac { x + 1 } { 2 } \right)$.
\item Find the gradient of the curve $y = \mathrm { f } ^ { - 1 } ( x )$ when $x = 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q8 [9]}}