Define Tab to be the mapping Ta$,$b : R $->$ R sending x $->$ ax $+$ b$.$ Here is an example of

how this function looks like for a $=2,$ b $=3$: T$2,3($x$)=2$x $+3.$

Define G :$=\{$Ta$,$b : a$,$ b in R and a $=0\}.$

$1.$ What is $($T$1,2$T$3,5)($x$)$ equal to$?$ Can you express the result of T$1,2$T$3,5$ as Ta$,$b for some a$,$ b in R$?$

$2.$ Prove that G forms a group under composition.

$3.$ Prove or disprove each subset below is a subgroup:

$($a$)$ GQ :$=\{$Ta$,$b : a$,$ b in Q$,$ a $=0\}.$

$($b$)$ GZ :$=\{$Ta$,$b : a$,$ b in Z$,$ a $=0\}.$

$($c$)$ H$1$ :$=\{$Ta$,$b : a $=1,$ b in R$\}.$

$($d$)$ H$2$ :$=\{$Ta$,$b : a$,$ b in R$,$ a $=$ b$,$ b $=0\}.$

$($e$)$ H$3$ :$=\{$Ta$,$b : a $=1,$ b in Z$\}.$ Explain each step in detail. Please don't do it on a computer programing language.