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 ${f{{\left({x}+{T}\right)}}}={f{{\left({x}\right)}}}$.