List of Notes - Category: Function Types

Increasing and Decreasing Functions

A function `f` is called increasing on interval `I` if `f(x_1)<f(x_2)` whenever `x_1<x_2` in `I`.

A function `f` is called decreasing on interval `I` if `f(x_1)>f(x_2)` whenever `x_1>x_2` in `I`.

Even Odd Function

If `f(x)=f(-x)` for every `x` in the domain of `f` then f is an even function.

For example, `f(x)=x^2` is even because for every `x` `f(-x)=(-x)^2=x^2=f(x)`.

If `f(-x)=-f(x)` for every `x` in the domain of `f` then `f` is an odd function.

Piecewise Function

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

For example, `f(x)={(1-x if x<0),(x^2 if x>=0):}` is piecewise because on interval `(-oo,0)` `f(x)=1-x` and on interval `[0,oo)` `f(x)=x^2`.

Periodic Function

Function `y=f(x)` is called periodic if exists such number `T!=0` that for any `x` from domain of the function `f(x+T)=f(x)`.

From the definition it follows that periodic function has infinitely many periods. If `T` is period of function then any number of the form `kT`, where `k` is integer,is also period of function.