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Absolute value function graph

  1. Oct 18, 2007 #1
    1. The problem statement, all variables and given/known data

    The problem is: Draw the graph of the following function:

    [tex]f(x)=|x+|x+|x-1|||[/tex]

    2. Relevant equations

    [tex]|x|=\left\{\begin{array}{cc}x,&\mbox{ if } x \geq 0\\-x,&\mbox{ if }x<0\end{array}\right[/tex]

    3. The attempt at a solution
    If the function were, for instance, [tex]g(x)=|x+1|-|x-1|[/tex], the solution wouldn't be a problem, because the two important points (x=-1 and x=1) can be recognized immediately, which implies analysing the three intervals ([tex]<-\infty,-1>[/tex] , [tex][-1,1>[/tex] , [tex][1,+\infty>[/tex]), and therefore the function g(x) can be seen as a compound of three different "sub-functions" on those intervals, ie:

    [tex]g(x)=\left\{\begin{array}{ll}
    g(x)=-2,&\mbox{ if } x \in <-\infty,-1>\\
    g(x)=2x,&\mbox{ if }x \in [-1,1>\\
    g(x)=2,&\mbox{ if }x \in [1,+\infty>\end{array}\right[/tex]

    and as such, its graph can be easily drawn.

    The same should be done for [tex]f(x)=|x+|x+|x-1|||[/tex]. But how? Where to start? If starting from the "inside", there would be, at the first step, two cases: [tex]x-1\geq 0[/tex] or [tex]x-1<0[/tex], which would lead to more sub-cases, so I'm not sure if this is the right approach to arrive at the graph of f(x).

    Any help would be much appreciated.
     
  2. jcsd
  3. Oct 19, 2007 #2
    Perhaps start by drawing x-1, then |x-1|, then x, then x+|x-1| etc.
    Or else the method you describe is the only one I can think of.
     
  4. Oct 21, 2007 #3
    First find the key points where the function changes. There are only 2 points where it changes. For example, plug in 10 for x.
    [tex]f(x)=|10+|10+|10-1|||[/tex]
    f(x) = 29
    the key points should not be that hard to find.
     
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