Are f(x) = x²/x and g(x) = x equal functions?

Step 1. (a). $\dfrac{x^2}{x}$ is not defined at $x = 0$, but $x$ is. The domains differ, so the functions are not equal, even though they agree wherever both exist.

Step 2. (b). Both are defined for every real $x$, and $\sqrt{x^2} = |x|$ always. They are equal.