Category: Function Types

Increasing and Decreasing Functions

A function $$${f{}}$$$ is called increasing on interval $$${I}$$$ if $$${f{{\left({x}_{{1}}\right)}}}<{f{{\left({x}_{{2}}\right)}}}$$$ whenever $$${x}_{{1}}<{x}_{{2}}$$$ in $$${I}$$$.

A function $$${f{}}$$$ is called decreasing on interval $$${I}$$$ if $$${f{{\left({x}_{{1}}\right)}}}>{f{{\left({x}_{{2}}\right)}}}$$$ whenever $$${x}_{{1}}>{x}_{{2}}$$$ in $$${I}$$$.

Even Odd Function

If $$${f{{\left({x}\right)}}}={f{{\left(-{x}\right)}}}$$$ for every $$${x}$$$ in the domain of $$${f{}}$$$ then f is an even function.

For example, $$${f{{\left({x}\right)}}}={{x}}^{{2}}$$$ is even because for every $$${x}$$$ $$${f{{\left(-{x}\right)}}}={{\left(-{x}\right)}}^{{2}}={{x}}^{{2}}={f{{\left({x}\right)}}}$$$.

Piecewise Function

When functions is determined by different formulas on different intervals then function is piecewise.

For example, $$${f{{\left({x}\right)}}}={\left\{\begin{array}{c}{1}-{x}{\quad\text{if}\quad}{x}<{0}\\{{x}}^{{2}}{\quad\text{if}\quad}{x}\ge{0}\\ \end{array}\right.}$$$ is piecewise because on interval $$${\left(-\infty,{0}\right)}$$$ $$${f{{\left({x}\right)}}}={1}-{x}$$$ and on interval $$${\left[{0},\infty\right)}$$$ $$${f{{\left({x}\right)}}}={{x}}^{{2}}$$$.

Periodic Function

Function $$$y={f{{\left({x}\right)}}}$$$ is called periodic if exists such number $$${T}\ne{0}$$$ that for any $$${x}$$$ from domain of the function $$${f{{\left({x}+{T}\right)}}}={f{{\left({x}\right)}}}$$$.