Domain is everything you're allowed to feed into the function machine; range is everything that can possibly come out. The maximal domain usually comes down to three common traps — a square root, a denominator, or a logarithm. On top of that, a question can state a domain, or the context can set one (time t ≥ 0). Pick a function below, then move x through the machine and watch exactly where it jams.
The maximal domain (every x the rule can take) comes from three common failure modes. Square roots need non-negative values under the radical, so you solve the inequality inside ≥ 0. Denominators can never equal zero, so you exclude every x that zeroes the bottom of a fraction — that's where vertical asymptotes live. Logarithms demand strictly positive arguments: zero is just as illegal as a negative. On the graph these traps show up as hard endpoints and walls the curve refuses to cross. When a question asks for the maximal domain, hunt for those three traps first, then write whatever survives in interval notation. A stated domain, or one set by the context (time t ≥ 0, a length > 0), or one restricted to make the function one-to-one, applies on top of that.
Domain is what the machine can eat, range is what it can spit out — and both must be written in interval notation or set-builder notation, using round brackets for any value the function approaches but never actually reaches.