\begin{enumerate}[label=(\alph*)]
\item Matthew is finding a formula for the inverse function $\text{arsinh } x$.
He writes his steps as follows:

Let $y = \sinh x$

$y = \frac{1}{2}(e^x - e^{-x})$

$2y = e^x - e^{-x}$

$0 = e^x - 2y - e^{-x}$

$0 = (e^x)^2 - 2ye^x - 1$

$0 = (e^x - y)^2 - y^2 - 1$

$y^2 + 1 = (e^x - y)^2$

$\pm \sqrt{y^2 + 1} = e^x - y$

$y + \sqrt{y^2 + 1} = e^x$

To find the inverse function, swap $x$ and $y$: $x + \sqrt{x^2 + 1} = e^y$

$\ln\left(x + \sqrt{x^2 + 1}\right) = y$

$\text{arsinh } x = \ln\left(x + \sqrt{x^2 + 1}\right)$

Identify, and explain, the error in Matthew's proof.
[2 marks]

\item Solve $\ln\left(x + \sqrt{x^2 + 1}\right) = 3$
[1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 1 2018 Q6 [3]}}