Domain is everything you're allowed to feed into the function machine; range is everything that can possibly come out. And every domain restriction on the course traces back to just three traps — a square root, a denominator, or a logarithm. Pick a function below, then drag x through the machine and watch exactly where it jams.
Domain restrictions are never arbitrary — they come from exactly three 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 an exam asks for the maximal domain, hunt for those three traps first, then write whatever survives in interval notation.
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.