The hyperbolic sine function is a one-to-one function, and thus has an inverse. As usual, we obtain the graph of the inverse hyperbolic sine function (also denoted by ) by reflecting the graph of about the line y=x:
Since is defined in terms of the exponential function, you should not be surprised that its inverse function can be expressed in terms of the logarithmic function:
Let's set , and try to solve for x:
This is a quadratic equation with instead of x as the variable. y will be considered a constant.
So using the quadratic formula, we obtain
Since for all x, and since for all y, we have to discard the solution with the minus sign, so
Read that last sentence again slowly!
We have found out that
Here it is: Express the inverse hyperbolic cosine functions in terms of the logarithmic function!