The unit step function is given in a combinatory form $θ(x+2)+θ(x+1)+θ(x)x$
, so what we have here is:
for $x< 0$:
$θ(x+2)=1 \ \ \ when \ \ -2<x$
$θ(x+1)=1 \ \ \ when \ \ -1< x$
$θ(x)x=0 $
Then for $x> 0$:
$θ(x+2)=1 \ \ \ when \ \ -2<x$
$θ(x+1)=1 \ \ \ when \ \ -1< x$
$θ(x)x=x $
So I thought we would get a straight horizontal line for $x<0$ at y=1, and then $y=x+1$ from $x>0$.
But wolfram plots this https://www.wolframalpha.com/input?i=%CE%B8%28x%2B2%29%2B%CE%B8%28x%2B1%29%2B%CE%B8%28x%29x
Why, and is my assumption not correct?
If we plot the Heaviside Unit Step, $\theta(x)$, we get
If we now plot $\theta(x+2)$, this is just shifting the previous by two to the left, we get
If we plot, $θ(x+2)+θ(x+1)$, notice how we get a unit step at $-2$, but also get one at $-1$
If we plot, $θ(x+2)+θ(x+1)+θ(x)x$, notice how we have the unit step at $-2$, then $-1$ and then $0$. Since it is $1$ at $x \ge 0$, notice how it is just the line $x$ to the right.