Let $F$ be a field, let $R_1=F[x]$ be the ring of polynomials with coefficients in $F$, and let $R_2$ be the ring of all functions from $F$ to itself, with addition and multiplication defined as the usual operations on functions with values in a ring. The function
$$\phi: R_1 \rightarrow R_2$$
which send the polynomial $f\in F[x]$ to the $F$-valued function $a \rightarrow f(a)$ on $F$ which induces, is a homomorphism.
When $F=\mathbb Q$ or $\mathbb R$, show that the homomorphism $\phi$ is injective but not surjective.
I would start like so
if $\phi(a) = b$ and $\phi(a') = b$, then $a=a'$. Since $a,a' \in F[x]$...