| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.2 This is a straightforward C3 question on functions requiring standard techniques: identifying range of an exponential function, finding an inverse (involving logarithms), and composing functions. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than the average A-level question. |
| 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 properties |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(x) > 0\) | B1 | |
| (b) \(y = 3e^{x-1}\), \(x - 1 = \ln\frac{y}{3}\) | M1 | |
| \(x = 1 + \ln\frac{y}{3}\) | M1 A2 | |
| \(f^{-1}(x) = 1 + \ln\frac{x}{3}\), \(x \in \mathbb{R}\), \(x > 0\) | ||
| (c) \(f(\ln 2) = 3e^{\ln 2 - 1} = 3e^{\ln 2} \cdot e^{-1} = 6e^{-1}\) | M1 A1 | |
| \(g(f(\ln 2)) = g(6e^{-1}) = 30e^{-1} - 2\) | A1 | |
| (d) \(f^{-1}g(x) = f^{-1}(5x-2) = 1 + \ln\frac{5x-2}{3}\) | M1 A1 | |
| \(\therefore 1 + \ln\frac{5x-2}{3} = 4\), \(\frac{5x-2}{3} = e^3\) | M1 | |
| \(x = \frac{1}{5}(3e^3 + 2)\) | A1 | (12) |
(a) $f(x) > 0$ | B1 |
(b) $y = 3e^{x-1}$, $x - 1 = \ln\frac{y}{3}$ | M1 |
$x = 1 + \ln\frac{y}{3}$ | M1 A2 |
$f^{-1}(x) = 1 + \ln\frac{x}{3}$, $x \in \mathbb{R}$, $x > 0$ |
(c) $f(\ln 2) = 3e^{\ln 2 - 1} = 3e^{\ln 2} \cdot e^{-1} = 6e^{-1}$ | M1 A1 |
$g(f(\ln 2)) = g(6e^{-1}) = 30e^{-1} - 2$ | A1 |
(d) $f^{-1}g(x) = f^{-1}(5x-2) = 1 + \ln\frac{5x-2}{3}$ | M1 A1 |
$\therefore 1 + \ln\frac{5x-2}{3} = 4$, $\frac{5x-2}{3} = e^3$ | M1 |
$x = \frac{1}{5}(3e^3 + 2)$ | A1 | **(12)**The function f is defined by
$$f : x \to 3e^{x-1}, \quad x \in \mathbb{R}.$$
\begin{enumerate}[label=(\alph*)]
\item State the range of f. [1]
\item Find an expression for $f^{-1}(x)$ and state its domain. [4]
\end{enumerate}
The function g is defined by
$$g : x \to 5x - 2, \quad x \in \mathbb{R}.$$
Find, in terms of e,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item the value of gf(ln 2), [3]
\item the solution of the equation
$$f^{-1}g(x) = 4.$$ [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q5 [12]}}