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Abstract Algebra

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Operations on Functions: Inverse of a Function

ABSALG-HSPXB1

Let $A$, $B$ and $C$ be sets, and let $g:A\rightarrow B$ and $f:B\rightarrow C$ be functions.

The composition $(f\circ g):A\rightarrow C$ is defined by $(f\circ g)(a)=f(g(a))$.

We say $g:A\rightarrow B$ has an inverse if and only if there is a function $f:B\rightarrow A$
such that $(f\circ g)(a)=a$, for all $a\in A$, and $(g\circ f)(b)=b$ for all $b\in B$.

Which of the following functions $g:\mathbb{R}\setminus\{0,1\}\rightarrow \mathbb{R}\setminus\{0,1\}$ have inverses?

Select ALL that apply.

A

$g(x)=x+\frac1x$

B

$g(x)=|x|$

C

$g(x)=2x+1$

D

$g(x)=1-x$

E

$g(x)=x^4$

F

$g(x)=x^3$