Function Fundamentals

A function is a rule that assigns to each input exactly one output. If $f$ maps a set $X$ to a set $Y$, we write $f: X \to Y$, and for an input $x$ we call $f(x)$ the image of $x$. The defining feature of a function is single-valuedness: every element of the domain has one and only one image. A relation that sends a single input to two different outputs — such as $y^2 = x$, which gives both $y = \sqrt{x}$ and $y = -\sqrt{x}$ — is not a function. Graphically this is captured by the vertical line test: any vertical line meets the graph of a function at most once.

Domain and range. The domain is the set of all permitted inputs; the range is the set of all outputs actually produced. When a domain is not stated, we take the largest set of real numbers for which the rule makes sense. Two restric