Let $\alpha :\scr{F}(\mathbb{R}) \rightarrow \mathbb{R} $ be defined by $\alpha(f)=f(1)$ and let $\beta :\scr{F}(\mathbb{R}) \rightarrow \mathbb{R} $ be defined by $\beta(f)=f(2)$.
Let $J$ be set of all functions from $\mathbb{R}$ to $\mathbb{R}$ whose graph passes through the point $(1,0)$ and let $K$ be set of all functions whose graph passes through $(2,0)$. Use the Fundamental Isomorphism Theorem to prove that $\mathbb{R} \cong {\scr F}(\mathbb{R})/J $ and $\mathbb{R} \cong {\scr F}(\mathbb{R})/ K$. Then conclude ${\scr F}(\mathbb{R})/ J \cong {\scr F}(\mathbb{R})/ K$.
I was able to prove that $\alpha,\beta$ were homomorphisms from $\scr{F}(\mathbb{R})$ onto $\mathbb{R}$ using the basic properties of the functions as well as the definitions of onto and homomorphisms, however the extension to this problem has managed to thoroughly confuse me.
Note that $J$ is the kernel of $\alpha$, so by the fundamental isomorphism theorem $\mathscr{F}(\mathbb{R})/J \cong \alpha[\mathscr{F}(\mathbb{R})]$. Here $\alpha[\mathscr{F}(\mathbb{R})]$ means the image of $\mathscr{F}(\mathbb{R})$ under $\alpha$. But assuming $\mathscr{F}(\mathbb{R})$ is the set of all functions on $\mathbb{R}$ this image is just $\mathbb{R}$, so $\mathscr{F}(\mathbb{R})/J \cong \mathbb{R}$.
By an analogous argument $\mathscr{F}(\mathbb{R})/K \cong \mathbb{R}$.
It immediately follows that $\mathscr{F}(\mathbb{R})/J \cong \mathscr{F}(\mathbb{R})/K$.