Determine type of function from points

Question

Assume $A$ is a set of points in $\mathbb{R^2}$. The points of $A$ were generated from a function that is either linear, polynomial, or exponential, but you don't know which. Is it possible to determine the function type from $A$ (though you don't need to determine the exact function itself)? If this is possible, must $|A| ≥ k$ for some $k \in \mathbb{R}$ in order to make this determination?