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Thread: function inverse

  1. #1
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    function inverse

    Can anyone find the inverse function of this:




    (will you please show work!)

    Thank you!
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  2. #2
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    Hi fruitkate21,

    We have $\displaystyle y=f(x)=\frac{10x-1}{2x+9}$ Now interchange $\displaystyle y$ and $\displaystyle x$ rearranging for $\displaystyle y$;

    $\displaystyle x=\frac{10y-1}{2y+9}$

    $\displaystyle \implies 2xy+9x=10y-1$

    $\displaystyle \implies 10y-2xy=9x+1$

    $\displaystyle \implies y(10-2x)=9x+1$

    $\displaystyle \implies y=f^{-1}(x)=\frac{9x+1}{10-2x}$
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  3. #3
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    Hello, fruitkate21!

    Find the inverse function of: .$\displaystyle f(x) \;=\;\frac{10x-1}{2x+9}$
    There are four basic steps . . .


    [1] Replace $\displaystyle f(x)$ with $\displaystyle y\!:\;\;y \;=\;\frac{10x-1}{2x+9}$


    [2] Interchange $\displaystyle x$'s and $\displaystyle y$'s: .$\displaystyle x \;=\;\frac{10y - 1}{2y + 9}$


    [3] Solve for $\displaystyle y\!:$

    . . .$\displaystyle x(2y+9) \:=\:10y - 1 $

    . . . $\displaystyle 2xy + 9x \;=\;10y - 1$

    . . $\displaystyle 2xy - 10y \;=\;\text{-}9x - 1$

    . . .$\displaystyle 2(x-5)y \;=\;\text{-}(9x + 1)$

    . . . . . . . .$\displaystyle y \;=\;\frac{\text{-}(9x+1)}{2(x-5)} $


    [4] Replace $\displaystyle y$ with $\displaystyle f^{\text{-}1}(x)\!:\;\;\boxed{f^{\text{-}1}(x) \;=\;\frac{1 + 9x}{2(5-x)}} $


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    If you don't have to show your work,
    . . there is a formula for this problem.


    If $\displaystyle f(x) \:=\:\frac{{\color{blue}a}x + b}{cx + {\color{blue}d}}$ . . . then: .$\displaystyle f^{-1}(x) \;=\;\frac{{\color{blue}d}x \:{\color{red}-}\: b}{{\color{red}-}cx+ {\color{blue}a}}$


    This is easily memorized:

    . . (1) Switch the numbers on the main diagonal
    . . . . .(upper-left and lower-right).

    . . (2) Change the signs on the other diagonal
    . . . . .(upper-right and lower-left).

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