Find inverse hyperbolic function

[synergix]

Homework Statement

find f^-1(t) and compute f^-1(0) where
f(t) = sinh(t)

Homework Equations

sinh(t)=( e^t- e^-t)/2

The Attempt at a Solution

t= (e^y- e^-y)/2

ln2t = y - - y

y = (ln2t)/2

but wikipedia says
http://upload.wikimedia.org/math/8/e/d/8edd21edb4cd413e462fe82ef4ac249b.png"

what silly mistake have i made this time?

 

[lanedance]

i would check the step where you take the logarithm

ony way that may work, would be to let y = f(t), x=e^t then try and solve for x(y) first

 

[Bohrok]

y = (e^t- e^-t)/2
2y = e^t- e^-t
2ye^t = e^2t - 1 (multiply both sides by e^t)

Let x = e^t, then you have a quadratic equation to help you isolate the t.

 

[synergix]

yes! this is how my instructor showed me but we have covered soo much this past couple of weeks I completely forgot about that technique. Thanks!

 

[synergix]

I used z instead of x and I got to
z^2-2tz-1=0
z= t+- sqrt(t^2+1)
z = e^y

so ln both sides

how do i know whether it is positive or negative?

since i can see from the formulas on wikipedia I went ahead with the rest and f-1(0) = ln1 = 0

 

[lanedance]

so is $t-\sqrt{t^2+1}$ greater or less than zero? is this for all t? consider what happens when you take the logarithm, and which is actually a solution to the orginal equation

also, I find the way you swap y & t is quite confusing
you start with:
$y(t) = sinh(t)$

so if I was you, I would solve for t(y)