| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.8 This question tests standard function theory at C3 level: stating the one-to-one condition, finding inverse functions, and function composition. All parts are routine textbook exercises requiring recall and direct application of standard techniques with no problem-solving insight needed. The multi-part structure adds length but not conceptual difficulty. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.06e Logarithm as inverse: ln(x) inverse of e^x |
\begin{enumerate}[label=(\alph*)]
\item Given that $f$ is a function,
\begin{enumerate}[label=(\roman*)]
\item state the condition for $f^{-1}$ to exist,
\item find $ff^{-1}(x)$. [2]
\end{enumerate}
\item The functions $g$ and $h$, are given by
$$g(x) = x^2 - 1,$$
$$h(x) = e^x + 1.$$
\begin{enumerate}[label=(\roman*)]
\item Suggest a domain for $g$ such that $g^{-1}$ exists.
\item Given the domain of $h$ is $(-\infty, \infty)$, find an expression for $h^{-1}(x)$ and sketch, using the same axes, the graphs of $h(x)$ and $h^{-1}(x)$. Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
\item Determine an expression for $gh(x)$ in its simplest form. [8]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 3 2018 Q12 [10]}}