Answer
a-
(i)one-to-one
(ii) not onto
b-
onto
Work Step by Step
$a-\\
f:\mathbb{Z}\rightarrow \mathbb{Z},f(n)=2-3n\\
A\,function\,\,f: \mathbb{Z} \rightarrow \mathbb{Z}\,is\,\,one-to-one\,\Leftrightarrow \\
\forall \,\,x_{1}\,and\,x_{2}\,\,in\,\,\mathbb{Z}\,\,if\,
f(x_{1}) = f(x_{2})\,\,then\,x_{1} = x_{2}.\\
so\,let\,f(x_{1})=f(x_{2})\\
\therefore 2-3x_{1}=2-3x_{2}\Leftrightarrow -3x_{1}=-3x_{2}\Leftrightarrow x_{1}=x_{2}\\
\therefore f\,is\,one-to-one\\
f\,is\,not\,onto\,as\,7\in \mathbb{Z}\,\\
is\,not\,image\,of\,some\,element\,in\,\mathbb{Z}\\
(because\,2-3n=7\Leftrightarrow -3n=5\Leftrightarrow n=\frac{-5}{3}\notin \mathbb{Z})\\
$
$b-\\
G:\mathbb{R}\rightarrow \mathbb{R},G(x)=2-3x\\
G: \mathbb{R} \rightarrow \mathbb{R} \,\,is\,\,onto\,\Leftrightarrow \,\\
\forall x\,in\,\mathbb{R} ,\exists y \in \mathbb{R}\,such\,that\, G(x) = y.\\
so\,let\,y\in \mathbb{R}\\
define\,\,x=\frac{2-y}{3}\\
by\,def.\,of\,G(x)\\
G(x)=2-3(\frac{2-y}{3})=2-2+y=y\\
so\,for\,any\,y\in \mathbb{R}\\
y\,is\,image\,of\,some\,element\,in\,\mathbb{R}
$