Monday, 25 March 2013

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 oftex2html_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
  • tex2html_wrap_inline81

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!
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