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Thread: Find Function expression

  1. #1
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    [Solved]Find Function expression

    The function f is defined by f(x) = x^3-x for -1=< x < 0 and f(x+1) = f(x) for all real x.
    Find an expression for f(x) for 0=< x < 1

    Dont worry I solved it...
    Last edited by Monster32432421; May 1st 2010 at 08:50 AM. Reason: I solved it..
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  2. #2
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    Hello, Monster32432421!

    You solved it? . . . Good for you!

    It's a strange problem . . .


    The function $\displaystyle f(x)$ is defined by: .$\displaystyle f(x) \:=\: x^3-x\;\text{ for }\text{-}1 \leq x < 0$

    and $\displaystyle f(x+1) \:=\: f(x)\;\text{ for all real }x.$

    Find an expression for $\displaystyle f(x)\:\text{ for }0 \leq x < 1$

    $\displaystyle f(x) \:=\:x^3-x$ has this graph:


    Code:
                      |
                  *   | 
               *    * |          *
             *       *|
                      |
        - - * - - - - * - - - - * - -
           -1         |         1
                      |*       *
           *          | *    *
                      |   *
                      |


    For the domain $\displaystyle [\text{-}1,0)$, we have this graph:


    Code:
                      |
                  *   |
               *    * |
             *       *|
                      |
        - - * - - - - * - - - -
           -1         |0
                      |



    Since $\displaystyle f(x+1) = f(x), $ the graph on $\displaystyle [0,1)$ is identical to the graph on $\displaystyle [\text{-}1,0)$

    The graph looks like this:


    Code:
                      |
                  *   |     *
               *    * |  *    *
             *       *|*       *
                      |
        - - * - - - - * - - - - * - -
           -1         |0         1
                      |


    So the graph of: $\displaystyle f_1(x) \:=\:x^3-x$ . is moved one unit to the right.

    We have: .$\displaystyle f_2(x) \;=\;(x-1)^3 - (x-1) \:=\:x^3 - 3x^2 + 2x$


    Therefore: .$\displaystyle f(x) \;=\;\begin{Bmatrix}x^3-x && \text{-}1 \leq x < 1 \\
    x^3-3x^2+2x && 0 \leq x < 1 \end{Bmatrix}$

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