Q.If $f(x) = x^2 - 1$ and $g(x) = x + 1$, find $\left(\dfrac{f}{g}\right)(x)$ in simplified form, stating its domain.

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. …