| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Moderate -0.3 This is a straightforward composite and inverse functions question with standard techniques. Part (a) requires simple substitution, part (b) is routine inverse function finding with logarithms, and part (c) combines these skills but follows a predictable method. All steps are textbook-standard with no novel insight 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 existence1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks |
|---|---|
| \(= g(5) = \log_2 16 = 4\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(y = \log_2(3x+1)\) | |
| \(3x + 1 = 2^y\) | M1 |
| \(x = \frac{1}{3}(2^y-1)\) | |
| \(g^{-1}(x) = \frac{1}{3}(2^y-1)\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(f \circ g^{-1}(x) = f[\frac{1}{3}(2^y-1)] = 2(2^y-1) - 1 = 2(2^y) - 3\) | M1 | |
| \(\therefore 2(2^y) - 3 = 2\) | ||
| \(2^y = \frac{5}{2}\) | A1 | |
| \(x = \frac{\ln 5 - \ln 2}{\ln 2}\) or \(\frac{\ln 5}{\ln 2}\) | M1 A1 | (9) |
**(a)**
$= g(5) = \log_2 16 = 4$ | M1 A1 |
**(b)**
$y = \log_2(3x+1)$ | |
$3x + 1 = 2^y$ | M1 |
$x = \frac{1}{3}(2^y-1)$ | |
$g^{-1}(x) = \frac{1}{3}(2^y-1)$ | M1 A1 |
**(c)**
$f \circ g^{-1}(x) = f[\frac{1}{3}(2^y-1)] = 2(2^y-1) - 1 = 2(2^y) - 3$ | M1 |
$\therefore 2(2^y) - 3 = 2$ | |
$2^y = \frac{5}{2}$ | A1 |
$x = \frac{\ln 5 - \ln 2}{\ln 2}$ or $\frac{\ln 5}{\ln 2}$ | M1 A1 | (9)
---5. The functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\
& \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Evaluate $\operatorname { gf } ( 1 )$.
\item Find an expression for $\mathrm { g } ^ { - 1 } ( x )$.
\item Find, in terms of natural logarithms, the solution of the equation
$$\mathrm { fg } ^ { - 1 } ( x ) = 2$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q5 [9]}}