Exercise 1.2.7 of Rudin in his book about real functions

I was studying chapter 1.2.7 of Rudin’s Real and Complex Analysis and was confused about one paragraph.
The author made a claim that:
Let $F$ be a field, $f_1,f_2\in F[x]$. The mapping $f_1\oplus f_2:F[x]\to F[x],g\to f_1(g)f_2(g)$, $f\in F[x]$, gives an isomorphism between $F[x]$ and $F[x]^2$ as $F$-vector spaces.
In order to show this claim, he said there are maps $f_1,f_2\in F[x]$ such that $f_1\oplus f_2$ is the identity map. Now, I don’t understand why? Why can we not simply take any two $f_1,f_2\in F[x]$? Am I missing something in the notation?

A:

(Answering the question “What?” instead of the question “Why?” which I find more confusing, I suppose.)
To see that the sum $f_1\oplus f_2$ is the identity map, it is sufficient to show that $f_1\oplus f_2$ sends the polynomial $x$ to its own constant $1$ coefficient. It is sufficient to show this for the monomial $x$, since the sum of the coefficients of any polynomial $g$ is the sum of the coefficients of $x$.

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