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Abstract Algebra

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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.

A

$f(x)=x^2$ and $g(x)=\sqrt{|x|}$

B

$f(x)=(x-1)^2$ and $g(x)=\sqrt{|x|}+1$

C

$f(x)=2x$ and $g(x)=x+1$

D

$f(x)=2x$ and $g(x)=\frac12 x$

E

$f(x)=x-1$ and $g(x)=\frac1{|x|+1}$