Functions: The Basics

A function is a machine. You put a number in, it does some arithmetic, and a number comes out. When you see $f(x)$, read it as “f of x.” It does not mean $f$ times $x$.

The one move is substitution. If a question asks for $f(3)$, replace every $x$ in the rule with $3$, then do the arithmetic. Exponents first, then multiply, then add or subtract.

Two things to watch. A negative number squared turns positive, so $(-3)^{2} = 9$. Absolute value bars mean distance from zero, so $|-5| = 5$.

A few extra facts show up again and again: $f(x) + 3$ shifts the graph up $3$ units, the vertex of $(x - h)^{2} + k$ sits at the point $(h, k)$, and a fraction is undefined wherever its bottom equals zero.