@Charles: Done.

@Igor: Yes Series would work but is kind of an overwork here. Also, finding $f_3$ is, in essence, the problem here.

@Pietro: Thanks! This is exactly the kind of solution I am looking for. Now assuming that $f_1$ and $f_2$ have the range $[0,1]$ and we want our family of combination's to lie in the same range, is there any way to ensure that?

@Charles: Yes I am interested in the parametric information and I just want a simple answer. Looking at functions as points in $L_2$3 does make it easy to visualize and may be also prove some existence results but is not helping in writing a parametrized version. @Mark: Getting an $F$ like you suggested in your first comment would be ideal. And I think there is one plane and one surface of a sphere (after doing a simple normalization), and we are interested in their intersection. After getting this intuition, geometry is no longer helpful to me. I really hope you could get something