I am looking for a particularly simple example of a regular function $f : V \to \mathbb{A}^1_k$ for some affine variety $V \subseteq \mathbb{A}^n_k$ over a field $k$, which cannot be expressed by a single rational function of $k(X_1, \dots, X_n)$.
Is there such an example if we choose $V = V(X^2+Y^2-1)$ to be the unit circle?
Added later: By "expressed" I mean "expressed as a map between sets". The converse question is: Do we find for any regular $f : V \to \mathbb{A}^1_k$ two polynomials $p,q \in k[X_1,\dots,X_n]$ such that $q$ has no zeros in $V$, and $f(x) = \frac{p(x)}{q(x)}$ holds for all $x \in V$?
Context: If we allow $V$ to be quasi-affine then there is such an example: Let $V = \{ (x,y) \in \mathbb{A}^2_k : x^2+y^2=1, (x,y) \neq (0,-1) \}$ and $f : V \to \mathbb{A}^1_k$ with $f(x,y) = \frac{1-y}{x}$ for $x \neq 0$ and $f(x,y) = \frac{x}{1+y}$ for $y \neq -1$. This function is well defined and regular. Of course the rational functions $\frac{1-y}{x}$ and $\frac{x}{1+y}$ represent the same element of $k(V)$ but I doubt that we can define $f$ by a single rational equation.
Am I understanding right from the comments that there is no analogous example in the case of a variety?