A question about ring homomorphisms

Homework Statement

If R is a domain with F=Frac(R), prove that Frac(R[x]) is isomorphic to F(x).

The Attempt at a Solution

Let $$\phi : Frac(R[x]) \rightarrow F(x)$$ be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that $$\phi$$ is a ring homomorphism. Let f,g,h,k be in R[x] such that f/h and g/k is in Frac(R[x]).

We know that

$$\phi(1,1) = 1/1= 1$$

$$\phi (fg, hk) = \frac{fg}{hk} = \frac{f}{h}\frac{g}{k} = \phi(f,h)\phi(g,k)$$

But I'm confused with the addition part...

$$\phi(f+g,h+k) = \frac{f+g}{h+k}$$

$$\phi(f,h)+\phi(g,k) = \frac{f}{h}+\frac{g}{k} = \frac{kf+gh}{h+k}$$

But now $$\phi(f+g,h+k) \not= \phi(f,h) + \phi(g,k)$$

Can anybody help with this?

Oh...ok I think I get what you mean...because $$\phi(f,h + g,k) = \phi(fk+gh, hk)$$, right?