How to graph $Abs(x) +Abs(x+1)$

Question

As I'm trying to graph this function on desmos which is a graphing calculator to verify my answer, there is a coordinate of (-0.5,1). The problem is I get every other whole coordinate EXCEPT that one despite trying the question multiple ways.

I'm mainly going about using the piecewise strategy

Answer 1

$f(x) = |x|+|x+1|=\begin{cases} -x -(x+1) & \forall x\in(-\infty,-1) \\-x +(x+1) &\forall x\in [-1,0) \\ x+x+1& \forall x\in [0,\infty) \end{cases} =\begin{cases} -2x -1 & \forall x\in(-\infty,-1) \\1 &\forall x\in [-1,0) \\ 2x+1& \forall x\in [0,\infty) \end{cases} $

So you'll get $1$ for all $x$ lying in the range $[-1,0]$

Answer 2

Here's the plot. What's the problem?

Answer 3

Good question.

• You're using the piecewise strategy, which sounds like you've noted that $\mathrm{abs}(x)$ behaves like the function $f(x)=x$ when $x>0$ and like the function $f(x)=-x$ when $x<0$.

• You also know that $\mathrm{abs}(x+1)$ behaves like the function $f(x)=x+1$ when $x+1 > 0$, and behaves like the function $f(x) = -(x+1)$ when $x+1 < 0$.

  (In other words, the two conditions are when $x > -1$ and when $x < -1$.)

• The conditions we've collected so far are: $x>0$ and $x<0$, and $x<-1$ and $x>-1$. This divides the space into three chunks: $$\qquad\qquad\qquad\leftarrow \underset{-1}{|} \qquad\qquad\qquad \underset{0}{|}\rightarrow \qquad\qquad\qquad $$

• Your function is $F(x) = \mathrm{abs}(x) + \mathrm{abs}(x-1)$.

  • In the first region, $x<-1$ and $x<0$, so it behaves like $F(x) = -x + -(x+1) = -2x-1$.
  • In the second region, $x>-1$ but $x<0$, so it behaves like $F(x) = x + -(x+1) = -1$.
  • In the third region, $x>-1$ and $x>0$, so it behaves like $F(x) = x + (x+1) = 2x+1$.

• The point $x=-0.5$ falls in the second region, because $x>-1$ and $x<0$. Therefore, for your function, $$F(-0.5) = (-0.5) + -(-0.5+1) = -0.5 - 0.5 = -1.$$