1.2 Basic classes of functions  (Page 6/28)

Algebraic functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function    is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function    is any function of the form $f\left(x\right)=p(x)\text{/}q\left(x\right),$ where $p(x)$ and $q(x)$ are polynomials. For example,

are rational functions. A root function    is a power function of the form $f\left(x\right)=x^{1\text{/}n},$ where $n$ is a positive integer greater than one. For example, $f(x)=x^{1\text{/}2}=\sqrt{x}$ is the square-root function and $g\left(x\right)=x^{1\text{/}3}=\sqrt[3]{x}$ is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, $f\left(x\right)=\sqrt{4-x^2}$ is an algebraic function.

The root functions $f\left(x\right)=x^{1\text{/}n}$ have defining characteristics depending on whether $n$ is odd or even. For all even integers $n\ge 2,$ the domain of $f\left(x\right)=x^{1\text{/}n}$ is the interval $[0,\infty ).$ For all odd integers $n\ge 1,$ the domain of $f\left(x\right)=x^{1\text{/}n}$ is the set of all real numbers. Since $x^{1\text{/}n}={\left(\text{−}\mathit{\text{x}}\right)}^{1\text{/}n}$ for odd integers $n,f\left(x\right)=x^{1\text{/}n}$ is an odd function if $n$ is odd. See the graphs of root functions for different values of $n$ in [link] .