Bounded Functions

Inverse FunctionsPeriodic Functions

A function is bounded on a set if its output values are confined between two horizontal lines. This concept captures the idea that the function cannot grow without limit on that set.

Quick Reference

Concept Meaning
bounded on such that fEM\lvert f(x) \rvertx \in EfEy = My = -MEffEMMYou can't use 'macro parameter character #' in math modesuch that</p> <mjx-container aria-label="|f(x)| < M \quad \text{for all } x \in E." class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="25.862ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 11431.1 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D453" xlink:href="#MJX-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(828,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1217,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1789,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2178,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(2733.8,0)"><use data-c="3C" xlink:href="#MJX-TEX-N-3C"></use></g><g data-mml-node="mi" transform="translate(3789.6,0)"><use data-c="1D440" xlink:href="#MJX-TEX-I-1D440"></use></g><g data-mml-node="mstyle" transform="translate(4840.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mtext" transform="translate(5840.6,0)"><use data-c="66" xlink:href="#MJX-TEX-N-66"></use><use data-c="6F" xlink:href="#MJX-TEX-N-6F" transform="translate(306,0)"></use><use data-c="72" xlink:href="#MJX-TEX-N-72" transform="translate(806,0)"></use><use data-c="20" xlink:href="#MJX-TEX-N-20" transform="translate(1198,0)"></use><use data-c="61" xlink:href="#MJX-TEX-N-61" transform="translate(1448,0)"></use><use data-c="6C" xlink:href="#MJX-TEX-N-6C" transform="translate(1948,0)"></use><use data-c="6C" xlink:href="#MJX-TEX-N-6C" transform="translate(2226,0)"></use><use data-c="A0" xlink:href="#MJX-TEX-N-A0" transform="translate(2504,0)"></use></g><g data-mml-node="mi" transform="translate(8594.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(9444.3,0)"><use data-c="2208" xlink:href="#MJX-TEX-N-2208"></use></g><g data-mml-node="mi" transform="translate(10389.1,0)"><use data-c="1D438" xlink:href="#MJX-TEX-I-1D438"></use></g><g data-mml-node="mo" transform="translate(11153.1,0)"><use data-c="2E" xlink:href="#MJX-TEX-N-2E"></use></g></g></g></svg></mjx-container><p>Otherwise,fEfEEy = My = -MM > 0f(x) = \lfloor x \rfloorxE = (-1, 1)x \in (-1, 1)\lfloor x \rfloor00 \leq x < 1-1-1 < x < 0You can't use 'macro parameter character #' in math mode). In either case: <mjx-container aria-label="|f(x)| \leq 1." class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="10.334ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4567.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D453" xlink:href="#MJX-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(828,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1217,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1789,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2178,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(2733.8,0)"><use data-c="2264" xlink:href="#MJX-TEX-N-2264"></use></g><g data-mml-node="mn" transform="translate(3789.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use></g></g></g></svg></mjx-container> Sof(-1, 1)M = 1M > 1f(x) = \lfloor x \rfloor\mathbb{R}Mn > M|f(n)| = n > Mf(x) = \lfloor x \rfloor(-1,1)\mathbb{R}g(x) = \dfrac{1}{x}g[1, \infty)\mathbb{R} - \{0\}[1, \infty)x \geq 10 < g(x) = 1/x \leq 1You can't use 'macro parameter character #' in math mode. Therefore: <mjx-container aria-label="|g(x)| \leq 1 \quad \text{for all } x \in [1, \infty)." class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="28.631ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 12654.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D454" xlink:href="#MJX-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(755,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1144,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1716,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2105,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(2660.8,0)"><use data-c="2264" xlink:href="#MJX-TEX-N-2264"></use></g><g data-mml-node="mn" transform="translate(3716.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mstyle" transform="translate(4216.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mtext" transform="translate(5216.6,0)"><use data-c="66" xlink:href="#MJX-TEX-N-66"></use><use data-c="6F" xlink:href="#MJX-TEX-N-6F" transform="translate(306,0)"></use><use data-c="72" xlink:href="#MJX-TEX-N-72" transform="translate(806,0)"></use><use data-c="20" xlink:href="#MJX-TEX-N-20" transform="translate(1198,0)"></use><use data-c="61" xlink:href="#MJX-TEX-N-61" transform="translate(1448,0)"></use><use data-c="6C" xlink:href="#MJX-TEX-N-6C" transform="translate(1948,0)"></use><use data-c="6C" xlink:href="#MJX-TEX-N-6C" transform="translate(2226,0)"></use><use data-c="A0" xlink:href="#MJX-TEX-N-A0" transform="translate(2504,0)"></use></g><g data-mml-node="mi" transform="translate(7970.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(8820.3,0)"><use data-c="2208" xlink:href="#MJX-TEX-N-2208"></use></g><g data-mml-node="mo" transform="translate(9765.1,0)"><use data-c="5B" xlink:href="#MJX-TEX-N-5B"></use></g><g data-mml-node="mn" transform="translate(10043.1,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(10543.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(10987.8,0)"><use data-c="221E" xlink:href="#MJX-TEX-N-221E"></use></g><g data-mml-node="mo" transform="translate(11987.8,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(12376.8,0)"><use data-c="2E" xlink:href="#MJX-TEX-N-2E"></use></g></g></g></svg></mjx-container> Sog[1, \infty)M = 1g(x) = 1/x[1, \infty)y = -1y = 1\mathbb{R} - \{0\}x \to 0^+g(x) \to +\inftyx \to 0^-g(x) \to -\inftyM|g(x)|x \neq 0gg(x) = 1/x\mathbb{R} - \{0\}x = 0h(x) = \dfrac{1}{x^2 + 1}h\mathbb{R}x \in \mathbb{R}You can't use 'macro parameter character #' in math mode: <mjx-container aria-label="x^2 + 1 \geq 1 \quad \text{(with equality at } x = 0\text{)}," class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="36.45ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 16111.1 1133.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2231,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(3008.8,0)"><use data-c="2265" xlink:href="#MJX-TEX-N-2265"></use></g><g data-mml-node="mn" transform="translate(4064.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mstyle" transform="translate(4564.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mtext" transform="translate(5564.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use><use data-c="77" xlink:href="#MJX-TEX-N-77" transform="translate(389,0)"></use><use data-c="69" xlink:href="#MJX-TEX-N-69" transform="translate(1111,0)"></use><use data-c="74" xlink:href="#MJX-TEX-N-74" transform="translate(1389,0)"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(1778,0)"></use><use data-c="20" xlink:href="#MJX-TEX-N-20" transform="translate(2334,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(2584,0)"></use><use data-c="71" xlink:href="#MJX-TEX-N-71" transform="translate(3028,0)"></use><use data-c="75" xlink:href="#MJX-TEX-N-75" transform="translate(3556,0)"></use><use data-c="61" xlink:href="#MJX-TEX-N-61" transform="translate(4112,0)"></use><use data-c="6C" xlink:href="#MJX-TEX-N-6C" transform="translate(4612,0)"></use><use data-c="69" xlink:href="#MJX-TEX-N-69" transform="translate(4890,0)"></use><use data-c="74" xlink:href="#MJX-TEX-N-74" transform="translate(5168,0)"></use><use data-c="79" xlink:href="#MJX-TEX-N-79" transform="translate(5557,0)"></use><use data-c="20" xlink:href="#MJX-TEX-N-20" transform="translate(6085,0)"></use><use data-c="61" xlink:href="#MJX-TEX-N-61" transform="translate(6335,0)"></use><use data-c="74" xlink:href="#MJX-TEX-N-74" transform="translate(6835,0)"></use><use data-c="A0" xlink:href="#MJX-TEX-N-A0" transform="translate(7224,0)"></use></g><g data-mml-node="mi" transform="translate(13038.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(13888.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(14944.1,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mtext" transform="translate(15444.1,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(15833.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g></g></g></svg></mjx-container> so <mjx-container aria-label="0 < h(x) = \frac{1}{x^2+1} \leq 1." class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.814ex;" xmlns="http://www.w3.org/2000/svg" width="23.474ex" height="4.851ex" role="img" focusable="false" viewBox="0 -1342 10375.7 2143.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(777.8,0)"><use data-c="3C" xlink:href="#MJX-TEX-N-3C"></use></g><g data-mml-node="mi" transform="translate(1833.6,0)"><use data-c="210E" xlink:href="#MJX-TEX-I-210E"></use></g><g data-mml-node="mo" transform="translate(2409.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2798.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(3370.6,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4037.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(5093.1,0)"><g data-mml-node="mn" transform="translate(1335.5,676)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-719.9)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2231,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g></g><rect width="2931" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(8541.9,0)"><use data-c="2264" xlink:href="#MJX-TEX-N-2264"></use></g><g data-mml-node="mn" transform="translate(9597.7,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use></g></g></g></svg></mjx-container> Therefore|h(x)| \leq 1x \in \mathbb{R}hM = 1|x| \to \inftyh(x) \to 0hh(0) = 1²h(x) = 1/(x^2 + 1)\mathbb{R}y = 0y = 1g(x) = 1/xh(x) = 1/(x^2+1)|f(x)| < Mxy = -My = Mg(x) = 1/x[1, \infty)\mathbb{R} - \{0\}Eh(x) = 1/(x^2+1)M = 11x = 00f(x) = cx|f(x)| = |c|M = |c| + 1|f(x)| < Mx$.
Inverse FunctionsPeriodic Functions
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