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 [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials. The following are some examples of rational functions.
A root function is a power function of the form [latex]f(x)=x^{1/n}[/latex], where [latex]n[/latex] is a positive integer greater than one. For example, [latex]f(x)=x^{1/2}=\sqrt{x}[/latex] is the square-root function and [latex]g(x)=x^{1/3}=\sqrt[3]{x}[/latex] is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, [latex]f(x)=\sqrt{4-x^2}[/latex] is an algebraic function.
algebraic functions
Algebraic functions are mathematical expressions combining constants and variables through operations like addition, multiplication, division, and taking roots. They encompass both rational functions, ratios of polynomials, and root functions, involving nth roots of the variable. Th
Transcendental Functions
Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions.
A trigonometric function relates the ratios of two sides of a right triangle. They are [latex]\sin x,\, \cos x, \, \tan x, \, \cot x,\, \sec x[/latex], and [latex]\csc x[/latex].
An exponential function is a function of the form [latex]f(x)=b^x[/latex], where the base [latex]b>0, \, b \ne 1[/latex].
A logarithmic function is a function of the form [latex]f(x)=\log_b(x)[/latex] for some constant [latex]b>0, \, b \ne 1[/latex], where [latex]\log_b(x)=y[/latex] if and only if [latex]b^y=x[/latex].
transcendental functions
Transcendental functions, including trigonometric, exponential, and logarithmic functions, are those which cannot be defined by a finite number of algebraic operations.
Classify each of the following functions, as algebraic or transcendental.
- [latex]f(x)= \dfrac{\sqrt{x^3+1}}{4x+2}[/latex]
- [latex]f(x)=2^{x^2}[/latex]
- [latex]f(x)=\sin (2x)[/latex]