Q.If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{4 - x}$, find the domain of $f + g$, $f - g$ and $fg$.

Just as numbers combine via $+,-,\times,\div$, two functions $f$ and $g$ combine pointwise into four new functions: $(f+g)(x)=f(x)+g(x)$, $(f-g)(x)=f(x)-g(x)$, $(f\cdot g)(x)=f(x)\cdot g(x)$, and $\left(\dfrac fg\right)(x)=\dfrac{f(x)}{g(x)}$ (this last one additionally requiring $g(x)\ne0$). Each combined function's domain is the intersection of $f$'s domain and $g$'s domain (the set of inputs valid for both simultaneously) — for the quotient, further shrunk to exclude any point where $g$ is zero.

In practice this means: to evaluate a combined function at a specific number, it is usually easiest to evaluate $f$ and $g$ separately at that number first, then combine the two results arithmetically (rather than building a combined formula and substituting). To find the domain of a combined function symbolically, find each piece's own domain restriction separately and intersect them.