How to deduce function $f$ given level sets at intersections with axes?
So given level set
$$f(x,y,z)=k$$
and its intersection points: $$f(x,0,0) \text{ at } (k^3,0,0)$$ $$f(0,y,0) \text{ at } (0,\text{some f(k) e.g. } 2k^2,0)$$ $$f(0,0,z) \text{ at } (0, 0, \text{some f(k)})$$
Then how can I deduce, what $f$ is?
Is $f$ necessarily unique?
EDIT: It seems like I misunderstood the question. Disregard.
Answering under the assumption that we're looking for all functions $f$ such that $f(x,0,0) = f^3(x,y,z)$ for all real $x,y,z$.
Fix $x$ and choose $y = z = 0$. So, $f(x,0,0) = f^3(x,0,0)$, and thus $f(x,0,0) \in \{-1, 0, 1\}$ for any $x$.
Now fix $x,y,$ and $z$. Note that $q = q^{\frac{1}{3}}$ for each $q \in \{-1, 0, 1\}$, so that means that $f(x,y,z) = f(x,0,0)$ for all $x,y,z$ reals. So, your function can be thought of as just a bunch of constant $yz$ planes, each of whose value is completely determined by the $x$ coord the plane passes through.
The only continuous functions that satisfy the rule are the 3 constant functions $f(x,y,z) = -1, 0,$ or $1$.