How do I graph the absolute value function with multiple layers?

[charm_quark]

Homework Statement

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

f(x)=|x+|x+|x-1|||

Homework Equations

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

The Attempt at a Solution

If the function were, for instance, g(x)=|x+1|-|x-1|, 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 (<-\infty,-1> , [-1,1> , [1,+\infty>), and therefore the function g(x) can be seen as a compound of three different "sub-functions" on those intervals, ie:

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

and as such, its graph can be easily drawn.

The same should be done for f(x)=|x+|x+|x-1|||. But how? Where to start? If starting from the "inside", there would be, at the first step, two cases: x-1\geq 0 or x-1&lt;0, 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.

 

[qspeechc]

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.

 

[MrXow]

First find the key points where the function changes. There are only 2 points where it changes. For example, plug in 10 for x.
f(x)=|10+|10+|10-1|||
f(x) = 29
the key points should not be that hard to find.