Two functions $$f$$, $$g$$ are defined on the set of real numbers $$\mathbb{R}$$, such that $$f(x) = |x|$$, $$g(x) = [x]$$, for all $$x \in \mathbb{R}$$. (where $$[x]$$ denotes the greatest integer less than or equal to $$x$$). Draw the graphs of the functions $$fog(x)$$, $$gof(x)$$.

PLACEHOLDER ANSWER - the answer is a pair of graphs, not a number. $$fog(x) = |[x]|$$, which is a step function taking the value $$|n|$$ on each interval $$[n, n + 1)$$; it equals 0 on $$[0, 1)$$, 1 on $$[1, 2)$$ and on $$[-1, 0)$$, 2 on $$[2, 3)$$ and on $$[-2, -1)$$, and so on. $$gof(x) = [|x|]$$ is an even step function equal to $$n$$ when $$n \le |x| < n + 1$$; it is 0 on $$(-1, 1)$$, 1 on $$[1, 2)$$ and $$(-2, -1]$$, and so on, symmetric about the $$y$$-axis.