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 tex2html_wrap_inline53 (also denoted by tex2html_wrap_inline55 ) by reflecting the graph of tex2html_wrap_inline57 about the line y=x:

Since tex2html_wrap_inline61 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 tex2html_wrap_inline63 , and try to solve for x:


This is a quadratic equation with tex2html_wrap_inline67 instead of x as the variable. y will be considered a constant.

So using the quadratic formula, we obtain


Since tex2html_wrap_inline73 for all x, and since tex2html_wrap_inline77 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 tex2html_wrap_inline83 .

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

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