Answer
a-
(i)one-to-one
(ii)not onto
b-
onto
Work Step by Step
$a-\\
f:\mathbb{Z}\rightarrow \mathbb{Z},f(n)=2n\\
A\,function\,\,f: X \rightarrow Y\,is\,\,one-to-one\,\Leftrightarrow \\
\forall \,\,x_{1}\,and\,x_{2}\,\,in\,\,X\,\,if\,
f(x_{1}) = f(x_{2})\,\,then\,x_{1} = x_{2}.\\
so\,let\,f(x_{1})=f(x_{2})\\
\therefore 2x_{1}=2x_{2}\Leftrightarrow x_{1}=x_{2}\\
\therefore f\,is\,one-to-one\\
f\,is\,not\,onto\,as\,7\in \mathbb{Z}\,\\
is\,not\,image\,of\,some\,element\,in\,\mathbb{Z}\\
(because\,7\,is\,odd\,number\,\,7\neq 2n)\\
b-\\
2\mathbb{Z} = \left \{ n \in \mathbb{Z}| n = 2k,\,for\,some\,integer\,k \right \}\\
h:\mathbb{Z}\rightarrow 2\mathbb{Z}\\
h(n)=2n\\
h: \mathbb{Z} \rightarrow 2\mathbb{Z} \,\,is\,\,onto\,\Leftrightarrow \,\\
\forall n\,in\,2\mathbb{Z} ,\exists n \in \mathbb{Z}\,such\,that\, h(x) = y.\\
so\,let\,n\in 2\mathbb{Z}\\
\Leftrightarrow n=2k (k\,is\,integer)\\
by\,def.\,of\,h(n)\\
n=2k=h(k)\\
so\,for\,any\,n\in 2\mathbb{Z}\\
n\,is\,image\,of\,some\,element\,in\,\mathbb{Z}(as\,k\in \mathbb{Z})$
