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Math Help - Finding the range of a function

  1. #1
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    Finding the range of a function

    Hi I'm doing an engineering course and i've got stuck on this question.

    f(x)= x/x^2-5x-6

    I've worked out the domain to be x is not = to -1, x is not equal to 6

    but I'm struggling to work out the range
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  2. #2
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    Re: Finding the range of a function

    \frac{x}{x^2}-5x-6=\frac{1}{x}-5x-6. Its domain is ℝ \ {0} and its range is ℝ.

    Finding the range of a function-graph2.png
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  3. #3
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    Re: Finding the range of a function

    thanks for your reply emakarov.

    everything sits under x as in

    x divide x^2-5x-6
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  4. #4
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    Re: Finding the range of a function

    The correct way to write this function is x / (x^2 - 5x - 6). I recommend reviewing the order of operations.

    It helps to sketch the graph of f(x) to come up with hypotheses to prove regarding the range.
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  5. #5
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    Re: Finding the range of a function

    Hello, atpod1!

    As emakarov pointed out, a graph would illustrate the behavior of this function.


    f(x) \:=\: \frac{x}{x^2-5x-6}

    I've worked out the domain to be: x \ne \text{-}1,\;x \ne =6.

    but I'm struggling to work out the range.

    There are vertical asymptotes at x = \text{-}1 and x = 6.


    Plot a few points and we find that:

    . . The graph contains the origin (0,0)

    . . To the far right and far left, the graph approaches the x-axis.

    . . To the left of x = \text{-}1, the graph is below the x-axis.

    . . On (\text{-}1,0), the graph is above the x-axis.

    . . On (0,6), the graph is below the x-axis.

    . . To the right of x = 6, the graph is above the x-axis.


    The graph looks like this:

    Code:
                          |
                    :     |     :
                    :*    |     :*
                    :     |     :
                    :     |     : *
                    : *   |     :  *
                    :     |     :    *
                    :  *  |     :       *
                    :   * |     :             *
      - - - - - - - + - - * - - + - - - - - - - - - -
          *       -1:     | *   :6
              *     :     |  *  :
                *   :     |     :
                 *  :     |   * :
                    :     |     :
                  * :     |     :
                    :     |    *:
                    :     |     :
                   *:     |     :
                    :      |     :
    It appears that f(x) will take on all values.

    Range: . (\text{-}\infty,\:\infty)
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  6. #6
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    Re: Finding the range of a function

    Thank you Soroban, your help is much appreciated
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