Rational points on schemes

This question was part of my homework I didn't turn in, but I'm curious
about how one could solve it.

X is a k-scheme, k being a field. f: k[x_1,...,x_n] -> O_X(X) a
homomorphism of k-algebras and f^*: X -> A^n the associated morphism.
Given a rational point x \in X(k), we want to show that f(x) =
(f_1(x),...,f_n(x)) where f_i = f(x_i) and f_i(x) is the image of f_i
in k[x_1,...,x_n] -> O_X(X) -> O_X,x -> k(x) = k, and the
identification A^n(k) = k^n.

I know that X(k) can be identified with the points in x for which k(x)
= k, and I also know that we can identify A^n(k) with k^n. However I
still get lost in all these identifications it seems and I can't show
that map above formally. I would be happy if someone could work out
some detailed description.

Regards

.

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