1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Trying to find an inverse equation - maybe cosh/sinh

  1. Sep 17, 2007 #1
    Given [tex]f(t) =[/tex] [tex]\frac{2e^{t} + 3e^{-t}}{e^{t} + 2e^{-t}}[/tex] find [tex]f^{-1}(t)[/tex]


    My attempt:
    i worked for about half an hour but i think im not doing the right thing. i tried multiplying it out, factorizing, pretty much just playing around with it.
    OMMGG mayb got it
     
    Last edited: Sep 17, 2007
  2. jcsd
  3. Sep 17, 2007 #2

    HallsofIvy

    User Avatar
    Science Advisor

    I would do this: write the equation as
    [tex]x= \frac{2e^t+ 3e^{-t}}{e^t+ 2e^{-t}}[/tex]
    Now swap x and t:
    [tex]t= \frac{2e^x+ 3e^{-x}}{e^x+ 2e^{-x}}[/tex]
    That "gives" the inverse function. Now you "only" have to solve for x!

    Multiply that denominator on both sides:
    [tex]te^x+ 2te^{-x}= 2e^x+ 3e^{-x}[/tex]
    Multiply the entire equation by ex
    [tex]te^{2x}+ 2t= 2e^{2x}+ 3[/tex]
    Let y= e^x so that you get a quadratic in y
    [tex]ty^2+ 2t= 2y^2+ 3[/tex]
    Solve that by a simple square root, then take ln of both sides to find x as a function of x.
     
  4. Sep 17, 2007 #3
    ooooohh thankszzz
    sooo [tex] x = ln\sqrt{\frac{3-2t}{t-2}}[/tex]
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...