# Absolute value function graph

1. Oct 18, 2007

### charm_quark

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

3. 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:

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

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 $$x-1<0$$, 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.

2. Oct 19, 2007

### 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.

3. Oct 21, 2007

### 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.