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: $(-\infty ,\infty )$ , Range is b, and the function is constant.

2) Identity function: f(x) = x

Properties: Domain = Range = $(-\infty ,\infty )$ 3) Square Function: $f(x)=x^{2}$ Properties: Domain = $(-\infty ,\infty )$ , while the range is $[0,\infty )$ . This function is also even.

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

5) Square Root Function: $f(x)={\sqrt {x}}$ Properties: The domain and range are both $[0,\infty )$ .

6) Cube Root Function: $f(x)={\sqrt[{3}]{x}}$ Properties: This function is odd and the domain = range = $(-\infty ,\infty )$ 7) Reciprocal Function: $f(x)={\frac {1}{x}}$ Properties: The domain is $(-\infty ,0)\cup (0,\infty )$ . The range is also the same.

8) Absolute Value Function: $f(x)=\left|x\right|$ Properties: Domain = $(-\infty ,\infty )$ , Range = $[0,\infty )$ . 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.