$[$Bonus$][8$ points$]$ Given two vectors x $=[$x$1,$ x$2]$ and y $=[$y$1,$ y$2],$ find a feature

mapping $()$ for each of the following functions, such that the function is equal to

the inner product between the mapped feature vectors, $($x$)$ and $($y$).$ For example,

$($x$$y$)0=($x$)($y$)$ where $($x$)=[1]$ and $($y$)=[1]$; $($x$$y$)1=($x$)($y$)$ where

$($x$)=$ x and $($y$)=$ y$.$

$($a$)[2$ points$]($x$$y$)2$

$($b$)[2$ points$]($x$$y$)3$

$($c$)[4$ points$]($x$$y$)$k where k is any positive integer.