Expressions of Inverse Hyperbolic functions [duplicate]

I have been trying to understand hyperbolic functions for some time now. I have a problem concerning the expressions of inverse hyperbolic functions. The text( G.B. Thomas ) mentions nothing about their expressions.

While plotting the Hyperbolic Sine was easy, plotting its inverse was not. By definition we have, $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$

I plotted its graph below- The dotted lines are the function's asymptotes. I then looked at the graph of inverse hyperbolic Sine and decided to obtain its expression. I tried by finding the inverses of the asymptotes because I knew that the inverse function would get close to them eventually. I plotted a graph. I was happy. But not for long. It turned out that I cannot have an algebraic combination of these two asymptotes. This was my second approach, to be honest. A holy approach to finding inverses is, express $x$ in terms of $y$. I got stuck at the very beginning and could never proceed. Is there a way around this? If there indeed is a way to express $x$ in terms of $y$, could anyone give me a few hints to do it? If there isn't, how do we obtain an expression for the inverse hyperbolic Sine?

marked as duplicate by Hans Lundmark, Arthur, kingW3, Glorfindel, NamasteAug 15 '17 at 20:24

• I will make sure to search for answers before posting a question. This link guides me well. – R004 Aug 15 '17 at 10:55
• A little search tip: use Google instead of the site's own search function. Like this: google.com/… – Hans Lundmark Aug 15 '17 at 10:58

Let $y\in R$
we look for $x\in R$ such that
$$e^x-e^{-x}-2y=0$$ or $$e^{2x}-2ye^x-1=0$$ put $t=e^x>0$ then $$t^2-2yt-1=0$$ hence $$t=e^x=y+\sqrt {1+y^2}$$
and finally $$x=\ln (y+\sqrt {y^2+1} )$$