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:

eqnarray14

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!

Click here to see the answer, and to continue.

Helmut Knaust
Fri Jul 19 11:01:21 MDT 1996