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Math Help - The range of this function?

  1. #1
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    Exclamation The range of this function?

    The function is f(x)= [6/(x+2)]-3

    I wanted to find the range, also could you please post your working as I want to understand how you came to your answer, thanks.

    Regards

    Ossy
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  2. #2
    Super Member Anonymous1's Avatar
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    f(x)= \frac{6}{x+2}-3

    Where is f(x) undefined? Naturally we look for such things as division by 0. When will we have division by 0? Clearly when x+2 = 0, i.e., when x= -2. So f(x) take on all values except when x = -2. What is your range?
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  3. #3
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    Quote Originally Posted by osmosis786 View Post
    The function is f(x)= [6/(x+2)]-3

    I wanted to find the range, also could you please post your working as I want to understand how you came to your answer, thanks.

    Regards

    Ossy
    one method of determining the range of a function is to determine the domain of the function's inverse. of course, this depends on whether the inverse exists and is relatively easy to find.

    y = \frac{6}{x+2} - 3

    swap variables ...

    x = \frac{6}{y+2} - 3

    solve for y ...

    x+3 = \frac{6}{y+2}

    y+2 = \frac{6}{x+3}

    y = \frac{6}{x+3} - 2

    this is the inverse function ... note that its domain is all real values x such that x \ne -3.

    so ... the range of the original function is all real y , y \ne -3.
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  4. #4
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    Hello, osmosis786!

    When possible, I like to sketch a graph.


    Find the range of: . f(x)\:=\:\frac{6}{x+2} -3

    The graph of . y \,=\,\frac{1}{x} .looks like this:
    Code:
                      |
                      |*
                      |
                      | *
                      |  *
                      |    *
                      |           *
        --------------+----------------
          *           |
                 *    |
                   *  |
                    * |
                      |
                     *|
                      |
    Domain: . x\,\neq\,0
    . Range: . y\,\neq\,0



    The graph of . y \:=\:\frac{1}{x+2}
    . . is the previous graph moved two units to the left.
    Code:
                  :   |
                  :*  |
                  :   |
                  : * |
                  :  *|
                  :   |*
                  :   |       *
    --------------+---+----------------
      *          -2   |
             *    :   |
               *  :   |
                * :   |
                  :   |
                 *:   |
                  :   |

    Domain: . x\,\neq\,-2
    . Range: . y\,\neq\,0



    The graph of . y \:=\:\frac{6}{x+2} . rises more "steeply"
    . . and "flattens slower", but has the same basic shape.



    The graph of . y \:=\:\frac{6}{x+2} - 3 . is the previous graph lowered 3 units.

    Code:
                  :   |
                  :*  |
                  :   |
                  : * |
    --------------+--*|--------------
                  :   |*
                  :   |       *
      - - - - - - o - | - - - - - -
      *        (-2,-3)|
             *    :   |
               *  :   |
                * :   |
                  :   |
                 *:   |
                  :   |

    Domain: . x\,\neq\,-2
    . Range: . y \,\neq\,-3

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