Is Lagrange interpolating polynomial unique on a region?

Question

Let's say I have some polynomial of degree $n$, call this $f(x)$ defined on a region [a,b].

If I find the Lagrange interpolating polynomial $P_{n}(x)$ given $n+1$ nodes from a to b, must $P_{n}(x) = f(x)$?

Answer 1

Yes. $P_n(x)$ is a polynomial of degree at most $n$ which agrees with $f$ at the $n+1$ points, and it is the only such polynomial.

$f(x)$ is a polynomial of degree at most $n$ wich agrees with $f$ at the $n+1$ points.

Therefore $P_n(x)$ and $f(x)$ must be identical.