| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.0 This is a standard C3 inverse function question requiring routine application of logarithm rules for part (a) and algebraic manipulation to find the inverse in part (b). Both parts follow textbook procedures with no novel problem-solving required, making it exactly average difficulty for A-level maths. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(2) = 2 + \ln 4\) | M1 A1 | |
| (b) \(y = 2 + \ln(3x - 2)\), \(3x - 2 = e^{y-2}\) | M1 | |
| \(f^{-1}(x) = \frac{1}{3}(2 + e^{x-2})\) | M1 A1 | (5 marks) |
**(a)** $f(2) = 2 + \ln 4$ | M1 A1
**(b)** $y = 2 + \ln(3x - 2)$, $3x - 2 = e^{y-2}$ | M1
$f^{-1}(x) = \frac{1}{3}(2 + e^{x-2})$ | M1 A1 | (5 marks)\begin{enumerate}
\item The function f is defined by
\end{enumerate}
$$\mathrm { f } ( x ) \equiv 2 + \ln ( 3 x - 2 ) , \quad x \in \mathbb { R } , \quad x > \frac { 2 } { 3 } .$$
(a) Find the exact value of $\mathrm { ff } ( 1 )$.\\
(b) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.\\
\hfill \mbox{\textit{Edexcel C3 Q1 [5]}}