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Math Help - absolute functions help

  1. #1
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    absolute functions help

    This problem was proposed to me recently and the absolute values are throwing me off.

    Given the functions f(x) = |-2x-4| and g(x) = |x+3|, considering appropriate domain restrictions, state the defining equations for the three regions of the function (f+g)(x).

    Obviously the solution will be a piecewise function, but I'm having trouble getting the equations for the three regions of the function(f+g)(x). Absolute signs can be tricky!
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  2. #2
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    Quote Originally Posted by skatefallen15 View Post
    This problem was proposed to me recently and the absolute values are throwing me off.

    Given the functions f(x) = |-2x-4| and g(x) = |x+3|, considering appropriate domain restrictions, state the defining equations for the three regions of the function (f+g)(x).

    Obviously the solution will be a piecewise function, but I'm having trouble getting the equations for the three regions of the function(f+g)(x). Absolute signs can be tricky!
    case 1 ...

    (-2x-4) \ge 0 and (x+3) \ge 0

    -2(x+2) \ge 0 ... x \ge -3

    x+2 \le 0

    x \le -2

    intersection is the interval [-3,-2]

    case 2 ...

    (-2x-4) \ge 0 and (x+3) < 0

    x \le -2 ... x < -3

    intersection is (-\infty,-3)

    case 3 ...

    (-2x-4) < 0 and (x+3) \ge 0

    -2(x+2) < 0 ... x \ge -3

    x+2 > 0

    x > -2

    intersection is (-2,\infty)

    case 4 ...

    (-2x-4) < 0 and (x+3) < 0

    x > -2 ... x < -3

    no intersection.


    so ... the three regions are (-\infty,-3) , [-3,-2] , and (-2,\infty)

    I'll leave you to define (f+g)(x) for each region.
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  3. #3
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    It is interesting to notice that |-2x-4|=|2x+4|.
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