A function is by definition a lookup table. The lookup table is just (x, f(x)). So, yes, trivially if you could construct the lookup table for f then you'd ap

A function is by definition a lookup table. The lookup table is just (x, f(x)). So, yes, trivially if you could construct the lookup table for f then you'd approximate f. But to construct it you have to know f. And to approximate it you need to know f at a dense set of points.