This question is regarding interpolation, say we are given table of data ,$x_0 <x_1<\dotsb<x_k<\dotsb<\dotsb<x_n$ as well as $f(x_0),f(x_1),\dotsc,f(x_n)$
it is said that
"We use forward difference formula for interpolating at values near top of table(since all forward differences are available) while we use backward difference formula for interpolating values near bottom of table(since all backward differences are available)."
This above statement looks problematic to me, since both backward difference formula, and forward difference formula give same polynomial for given table of data. then we should be able to interpolate any point $x_k$ within $x_0 $and $x_n$ by either backward difference formula, or forward difference formula regardless of whether it is near beginning or end of table.
OR more precisely, my question is:
By using forward or backward difference formulae to find $f(x_k)$, where $x_0<x_k<x_n$ , the result is different, although the polynomial $f(x)$ obtained from these two is same. Why?