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Thread: inverse functions

  1. #1
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    Unhappy inverse functions

    how do you find the inverse of f(X)=2^ -x ?

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  2. #2
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    Hello, ericchung912!

    How do you find the inverse of: .$\displaystyle f(x)\:=\:2^{-x}$ ?

    We have: . $\displaystyle f(x) \;=\;2^{-x}$

    Replace $\displaystyle y$ with $\displaystyle f(x)\!:\;\;y \;=\;2^{-x}$

    Switch x's and y's: .$\displaystyle x \;=\;2^{-y}$


    Solve for y: .$\displaystyle 2^{-y} \;=\;x$

    Take logs: .$\displaystyle \ln\left(2^{-y}\right) \;=\;\ln(x) \quad\Rightarrow\quad -y\!\cdot\!\ln(2) \;=\;\ln(x) \quad\Rightarrow\quad y \;=\;-\frac{\ln(x)}{\ln(2)} $

    Therefore: .$\displaystyle f^{-1}(x) \;=\;-\frac{\ln(x)}{\ln(2)} $

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