Properties of the Range — Cheat sheet

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Key method

Useful methods include completing the square, monotonicity on the stated domain, and solving $y=f(x)$ for an allowed $x$ . Endpoint inclusion follows from whether an input actually attains the candidate output.

Example

For $f(x)=(x-2)^2+3$ on $\mathbb R$ , the square gives $f(x)\ge3$ . Conversely, for any $y\ge3$ , choose $x=2+\sqrt{y-3}$ , which produces $f(x)=y$ . The range is exactly $[3,\infty)$ . For $1/x$ , output zero is impossible, while every nonzero $y$ is attained by $x=1/y$ .

Avoid this mistake

A graph window can suggest a range but cannot prove behavior outside its visible portion.