# Library of Functions

## Introduction

The Library of Functions is a collection of functions for which you should know the basic properties and shape of graph. The section is finished off with a discussion of piecewise-defined functions.

## Library of Functions

1) Constant Function: Constant functions have the form f(x) = b where b is a real number.

Properties: Domain: ${\displaystyle (-\infty ,\infty )}$, Range is b, and the function is constant.

2) Identity function: f(x) = x

Properties: Domain = Range = ${\displaystyle (-\infty ,\infty )}$

3) Square Function: ${\displaystyle f(x)=x^{2}}$

Properties: Domain = ${\displaystyle (-\infty ,\infty )}$, while the range is ${\displaystyle [0,\infty )}$. This function is also even.

4) Cube Function: ${\displaystyle f(x)=x^{3}}$

Properties: Domain = Range = ${\displaystyle (-\infty ,\infty )}$. ${\displaystyle x^{3}}$ is an odd function too.

5) Square Root Function: ${\displaystyle f(x)={\sqrt {x}}}$

Properties: The domain and range are both ${\displaystyle [0,\infty )}$.

6) Cube Root Function: ${\displaystyle f(x)={\sqrt[{3}]{x}}}$

Properties: This function is odd and the domain = range = ${\displaystyle (-\infty ,\infty )}$

7) Reciprocal Function: ${\displaystyle f(x)={\frac {1}{x}}}$

Properties: The domain is ${\displaystyle (-\infty ,0)\cup (0,\infty )}$. The range is also the same.

8) Absolute Value Function: ${\displaystyle f(x)=\left|x\right|}$

Properties: Domain = ${\displaystyle (-\infty ,\infty )}$, Range = ${\displaystyle [0,\infty )}$. This function is also even.

## Piecewise-defined Functions

A piecewise-defined function is a function that takes on the values of different functions depending on the x-value.