Easy# Operations on Functions: Noncommutativity of Composition

ABSALG-RPKBW1

For two functions $f, g$ such that the image of $g$ is contained in the domain of $f$, define the composition $f\circ g$ of $f$ and $g$ by $(f\circ g)(x)=f(g(x))$, for all $x$ in the domain of $g$.

In this question, we assume $x$ takes values in $\mathbb{R}$, and we denote by $|x|$ the absolute value of $x\in\mathbb{R}$.

For $x\ge 0$, we always assume that $\sqrt{x}\ge0$. That is, we choose the nonnegative square root.

For which of the following pairs of functions $g, f$ do we have $g\circ f=f\circ g$?

Select **ALL** that apply.