**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

#### 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!