Inverse Hyperbolic Functions

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

and consequently

Read that last sentence again slowly!
We have found out that

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Try it yourself!

You know what's coming up, don't you? Here's the graph. Note that the hyperbolic cosine function is not one-to-one, so let's restrict the domain to  .


Here it is: Express the inverse hyperbolic cosine functions in terms of the logarithmic function!