Function Notation and Evaluating at a Value (F 401, F 501)

f(x) is a label, not a product. The letter f names the machine, x is the slot for the input, and f(x) is the output that comes out the other side. When you see f(3), that's an instruction: take the rule for f and put 3 everywhere x appears. So if f(x) = 2x² - 5x + 4, then f(3) means 2(3)² - 5(3) + 4, not f times 3.

The same move works even when the input isn't a bare number. For f(a + 2), you still replace every x in the rule with the whole expression a + 2, parentheses and all, then expand and combine like terms. This is where the test likes to trip people up: it will ask you to evaluate at x + 1, x - 3, or some other expression instead of a clean number, to check that you understand substitution as a process rather than a lookup table.

This idea is the base layer for everything else in the unit. Once you can substitute an expression into a rule, you can read any of the four representations of a function, find domain and range by asking what inputs and outputs are allowed, spot vertical asymptotes as places the rule breaks down, and handle two-input or made-up-operation notation, which is just this same substitution trick with more slots to fill.