Prove that $\alpha$ and $\beta$ are homomorphisms from $F(\mathbb{R})$ onto $\mathbb{R}$ Let $\alpha : F(\mathbb{R}) \to \mathbb{R}$ be defined by $\alpha(f)=f(1)$ and let $\beta : F(\mathbb{R}) \to \mathbb{R}$ be defined by $\beta(f)=f(2)$
Prove that $\alpha$ and $\beta$ are homomorphisms from $F(\mathbb{R})$ onto $\mathbb{R}$.
I'm having trouble with this question because it is a bit confusing to me. I know homomorphisms are defined by a function and a property. In this particular case, the function is known which is $\alpha$ and $\beta$ but showing the property of homomorphism is confusing. Do I treat each separately or as one?
Meaning, do I need to show:
$$\alpha(fg)=f(1)g(1) \text{ where } f,g \in F(\mathbb{R})$$
or do I need to show
$$\alpha(f)\beta(f)=f(1)f(2)?$$
 A: The former. That is, you treat $\alpha$ and $\beta$ separately, they do not interact.
To show that $\alpha$ is a homomorphism, you need to show that $\alpha(fg) = \alpha(f)\alpha(g)$.
To show that $\beta$ is a homomorphism, you need to show that $\beta(fg) = \beta(f)\beta(g)$.
Think of this as a two part question where part one asks you to prove that $\alpha$ is a homomorphism and part two asks you to prove that $\beta$ is a homomorphism.
Added Later: I just noticed that you are not just asked to show that $\alpha$ and $\beta$ are homomorphisms from $F(\mathbb{R})$ to $\mathbb{R}$, but homomorphisms from $F(\mathbb{R})$ onto $\mathbb{R}$. So in addition to showing that $\alpha$ and $\beta$ are homomorphisms, you must also show that they are onto.
A: Imagine  functions as a huge object that nobody can see them in full. And $\alpha$ is a watchman at $1$, ( $\beta$ a watch man at 2 etc). You ask these guys any info about functions they can only give the value of the function at $1$. (and at 2 for $\beta$).
Now if you know $h=fg$, (otherwise nothing about the functions) and $\alpha$ gives you the values for $f$ and $g$ cannot you predict what $\alpha$ would say about $h$? (and similarly for $k=\alpha+\beta$?).
