Quick questions on Sketching y = f(x) +/- g(x) by adding ordinates

The richest payoff of adding ordinates is spotting an oblique asymptote, a slanted line the curve approaches. Take the hyperbola $f(x) = \dfrac{2}{x}$ and the line $g(x) = x$. Their sum is $s(x) = x + \dfrac{2}{x}$.

Absolute-value graphs are made of straight segments, and a fact worth pinning is that the sum of two linear pieces is linear. So when you add two absolute-value graphs, the result is again made of straight-line sections, with corners only where one of the originals bends. Take $f(x) = |x + 2|$, which bends at $x = -2$, and $g(x) = |x - 1|$, which bends at $x = 1$. The sum $s(x) = |x + 2| + |x - 1|$ has corners at those two $x$-values and is straight everywhere else.

You can only add two heights where both functions are defined, so the domain of $s(x) = f(x) + g(x)$ (and of the difference) is the intersection of the two domains. For example, $f(x) = \sqrt{x}$ (domain $x \ge 0$) plus $g(x) = \dfrac{1}{x}$ (domain $x \ne 0$) gives a sum defined only for $x > 0$. Never extend the sum into a region where one of the originals does not exist.