Understand a function as a rule assigning each input exactly one output; use f(x) notation and read graphs.
A function is a machine: numbers go in, a rule acts, numbers come out. The vending-machine law: each button gives exactly ONE item. If the same input could give two different outputs, it isn't a function. We write f(x) = 2x + 1, read "f of x" — f is the machine's name, x is what you feed it. f(4) means "feed it 4": f(4) = 2(4) + 1 = 9.
A graph is the machine's complete résumé: every point (x, y) says "input x came out as y." That's why the vertical line test works — a vertical line crossing the graph twice would mean one input with two outputs, which the law forbids.
Mastery looks like: They evaluate f(anything), solve f(x) = k backwards, and explain the one-output law in their own words.
Common stumbles: Reading f(x) as "f times x"; mixing up which axis is input; forgetting a graph point IS an input-output pair.