KöMaL 2022 A821

$a)$ Is it possible to find a function $f\colon \mathbb{N}^2\to \mathbb{N}$ such that for every function $g\colon \mathbb{N}\to \mathbb{N}$ and positive integer $m$ there exists $n \in \mathbb{N}$ such that set $\big\{k\in \mathbb{N} : f(n,k)=g(k)\big\}$ has at least $m$ elements? $b)$ Is it possible to find a function $f\colon \mathbb{N}^2\to \mathbb{N}$ such that for every function $g\colon \mathbb{N}\to \mathbb{N}$ there exists $n \in \mathbb{N}$ such that set $\big\{k\in \mathbb{N} \colon f(n,k)=g(k)\big\}$ has an infinite number of elements?