I'm going through "Calculus" by Tom Apostol. And I'm in this section:
I think the book assumes that from the example I can extrapolate how the graph for any addition of step functions is done; nonetheless, I don't understand that example. So a problem arises now that I have to do the first exercise.
So, when I'm going to do $a)$ I know how to graph $\lfloor x \rfloor$ and $\lfloor 2x \rfloor$, I even know how to do the common refinement, but not the graph of $\lfloor 2x \rfloor$+$\lfloor x \rfloor$ itself.
So, do you think you can tell me how step functions are added and multiplied? thanks in advance.
Best Answer
Looks like you have the right idea. You divide the domain into sub-intervals, and evaluate in each sub-interval.
$f(x) + g(x) = \begin {cases} -3-6 = 7 & -3 \le x < -2.5\\-3-5 = 6 & -2.5 \le x < -2\\-2-4 = 5 & -2 \le x < -1.5\\&\vdots\end{cases}$
$f(x)g(x) = \begin {cases} (-3)(-6) = 18 & -3 \le x < -2.5\\(-3)(-5) = 15 & -2.5 \le x < -2\\(-2)(-4) = 8 & -2 \le x < -1.5\\&\vdots\end{cases}$