Matthew is finding a formula for the inverse function \(\operatorname { arsinh } x\).
He writes his steps as follows:
$$\begin{gathered}
\text { Let } y = \sinh x
y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)
2 y = \mathrm { e } ^ { x } - \mathrm { e } ^ { - x }
0 = \mathrm { e } ^ { x } - 2 y - \mathrm { e } ^ { - x }
0 = \left( \mathrm { e } ^ { x } \right) ^ { 2 } - 2 y \mathrm { e } ^ { x } - 1
0 = \left( \mathrm { e } ^ { x } - y \right) ^ { 2 } - y ^ { 2 } - 1
y ^ { 2 } + 1 = \left( \mathrm { e } ^ { x } - y \right) ^ { 2 }
\pm \sqrt { y ^ { 2 } + 1 } = \mathrm { e } ^ { x } - y
y \pm \sqrt { y ^ { 2 } + 1 } = \mathrm { e } ^ { x }
\end{gathered}$$
To find the inverse function, swap \(x\) and \(y : x \pm \sqrt { x ^ { 2 } + 1 } = \mathrm { e } ^ { y }\)
$$\begin{gathered}
\ln \left( x \pm \sqrt { x ^ { 2 } + 1 } \right) = y
\operatorname { arsinh } x = \ln \left( x \pm \sqrt { x ^ { 2 } + 1 } \right)
\end{gathered}$$
Identify, and explain, the error in Matthew's proof.
6
Solve \(\ln \left( x + \sqrt { x ^ { 2 } + 1 } \right) = 3\)