Calculate a CDF given two PDFs. I have two PDF's, $f_1(x)$ and $f_2(x)$ and I need to find the CDF (not PDF) of $f_3(x)$ where    
$f_3(x) = \frac{1}{2} (f_1(x) + f_2(x))$  
I have already calculated $F_1$ and $ F_2$, the CDF's of $f_1$ and $f_2$ respectively. 
Can I simply take half the sum of the two CDF's as follows
$F_3(x) = \frac{1}{2}( F_1(x) + F_2(x))$   
or do I instead need to compute $f_3$ and then calculate $F_3$ from there? 

You can simply take the average of the CDF's, because the CDF is found by taking an integral, which is  a linear operator.  Scale factors like $1/2$ can "go outside the integral sign", and so can summation.
If we have some integral transform $T$ such that $T(f(x))$ takes some integral over $f(x)$ to produce another function $F(x)$, it will be a linear operator. This means that you can rely on equivalences such as $T(g(x) + h(x)) = T(g(x)) + T(h(x))$ and $T(C\cdot g(x)) = c\cdot T(g(x))$.
