Sketch the graph of $\sqrt{y^2-1}=\dfrac{1}{2}\left(e^x-e^{-x}\right)$ and also of $\sqrt{y^2+1}=\dfrac{1}{2}\left(e^x+e^{-x}\right)$. In each case find an explicit formula for $x$ in terms of $y$.
This is part of an admissions quiz, so unfortunately no access to Desmos or similar. It looks like hyperbolic functions might be involved.
We get from $\sqrt{y^2-1}=\dfrac{1}{2}\left(e^x-e^{-x}\right)$:
$y^2=\cosh^2(x)$. Hence $y=\pm \cosh(x)$.