Results 1 to 2 of 2

Thread: [SOLVED] Formula for inverse function

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    18

    [SOLVED] Formula for inverse function

    Let f(x)=20/(1+3e^-x). Find a formula for f^-1(x).

    So I tried to do it this way:

    y=f(x)=20/(1+3e^-x).

    x=20/(1+3e^-y)

    20/x=1+3e^-y

    log ((20/x)-1)/log(3e)

    Does this look right, or am I going about this in the wrong way?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, CalcQueen!

    You were okay up to the last step(s) (which you didn't show).


    Let $\displaystyle f(x)\,=\,\frac{20}{1+3e^{-x}}$ . Find $\displaystyle f^{-1}(x)$

    So I tried to do it this way:

    . . $\displaystyle y\:=\:\frac{20}{1+3e^{-x}}$

    . . $\displaystyle x\:=\:\frac{20}{1+3e^{-y}}$

    . . $\displaystyle 1+3e^{-y} \:=\:\frac{20}{x}$ . . . . good!

    . . $\displaystyle \frac{\log\left (\frac{20}{x}-1\right)}{\log(3e)}$ . ?

    You had: .$\displaystyle 1 + 3e^{-y} \:=\:\frac{20}{x}$

    Then: .$\displaystyle 3e^{-y} \:=\:\frac{20}{x} - 1 \quad\Rightarrow\quad 3e^{-y}\:=\:\frac{20-x}{x} \quad\Rightarrow\quad e^{-y} \:=\:\frac{20-x}{3x}$

    Take logs: .$\displaystyle \ln\left(e^{-y}\right) \:=\:\ln\left(\frac{20-x}{3x}\right) \quad\Rightarrow\quad (-y)\ln(e) \:=\:\ln\left(\frac{20-x}{3x}\right)$


    Since $\displaystyle \ln(e) = 1$, we have: .$\displaystyle -y \:=\:\ln\left(\frac{20-x}{3x}\right) $

    Finally: .$\displaystyle y \:=\:-\ln\left(\frac{20-x}{3x}\right) \quad\Rightarrow\quad y \:=\:\ln\left(\frac{20-x}{3x}\right)^{-1} \quad\Rightarrow\quad\boxed{ y \:=\:\ln\left(\frac{3x}{20-x}\right)}$

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] inverse of a function
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: Feb 11th 2010, 11:23 AM
  2. Replies: 5
    Last Post: Jan 30th 2010, 07:18 PM
  3. [SOLVED] Inverse Function
    Posted in the Calculus Forum
    Replies: 7
    Last Post: Jan 14th 2010, 08:03 AM
  4. [SOLVED] inverse of an ln function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 28th 2009, 10:58 AM
  5. [SOLVED] inverse of a function
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Feb 18th 2009, 07:47 PM

Search Tags


/mathhelpforum @mathhelpforum