Domain and Range — Cheat sheet

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Domain restrictions at a glance

ExpressionRequirement
1g(x)\frac{1}{g(x)}g(x)≠0g(x) \ne 0
g(x)2n\sqrt[2n]{g(x)}g(x)≥0g(x) \ge 0
ln⁡g(x)\ln g(x), log⁡ag(x)\log_a g(x)g(x)>0g(x) > 0
tan⁡g(x)\tan g(x)g(x)≠π2+kπg(x) \ne \frac{\pi}{2} + k\pi

Range tactics

  • Quadratics: complete the square; vertex gives the extreme.
  • ⋅\sqrt{\cdot} outputs are ≥0\ge 0; axa^x outputs are >0> 0.
  • When stuck: solve y=f(x)y = f(x) for xx; the yy that admit a solution form the range.