For a field K, we always have K(x)\subseteq K(x,y). If y\in K(x), it seems to be true that K(x,y)\subseteq K(x). My line of reasoning is this: any element h of K(x,y) is a quotient of polynomials with coefficients in K and "variables" x,y\in K(x). Each of these polynomials (in the 'numerator' and 'denominator' of h) is a polynomial in K[x]. Therefore being a field, K(x) must contain h, so K(x,y)\subseteq K(x). (i.e. we have equality in this case, so K(x) is an extension field of K(y)).
Alternatively K(x,y)=K(x)(y) is the smallest subfield of K containing K(x) and y, which is K(x) if y\in K(x).
Do these explanations make sense??
Thanks in advance.