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Radical Functions
An expression with roots is called a radical function, there are many kinds of roots, square root and cube root being the most common.
Learning Objective

Discover how to graph radical functions by examining the domain of the function
Key Points
 Roots are the inverse operation for exponents. If
$\sqrt [ n ]{ x } = r$ then${r}^{n}=x$ .  If the square root of a number is taken, the result is a number which when squared gives the first number.
 The cube root is the number which, when cubed, or multiplied by itself and then multiplied by itself again, gives back the original number.
 If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places.. Such a number is described as irrational.
Terms

root
the number which,when plugged into the equation, will produce a zero.

radical
A root (of a number or quantity).
Full Text
Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance
If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401?
Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When graphing functions, expressions that use exponentiation use curves instead of lines. Using algebra will show that not all of these expressions are functions and that knowing when an expression is a relation or a function allows certain types of assumptions to be made. These assumptions can be used to build mental models for topics that would otherwise be impossible to understand.
For now, deal with roots by turning them back into exponents. If a root is defined as the
Square root
If the square root of a number is taken, the result is a number which when squared gives the first number. This can be written symbolically as:
In the series of real numbers
Cube roots
Roots do not have to be square. The cube root of a number (
Other roots
There are an infinite number of possible roots all in the form of
Graphs of Radical Functions
Since roots are simply the inverse of exponents, graphing roots can be seen as just graphing exponents with the axes reversed. The shape of the radical graph will resemble the shape of the related exponent graph it were rotated 90degrees clockwise. For example, the graph of
Irrational numbers
If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places. Such a number is described as irrational and is defined as a number which cannot be written as a rational number:
The result of taking the square root is written with the approximately equal sign because the result is an irrational value which cannot be written in decimal notation exactly. Writing the square root of 3 or any other nonsquare number as
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Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 e
 Appears in these related concepts: Derivatives of Exponential Functions, Derivatives of Logarithmic Functions, and The Natural Exponential Function: Differentiation and Integration
 exponent
 Appears in these related concepts: Logarithmic Functions, Solving General Problems with Logarithms and Exponents, and Scientific Notation
 exponentiation
 Appears in these related concepts: Integer Exponents, Logarithmic Functions, and Simplifying Expressions of the Form log_a a^x and a(log_a x)
 expression
 Appears in these related concepts: Simplifying, Multiplying, and Dividing, Compound Inequalities, and Expressions and Sets of Numbers
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Average Value of a Function
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Equations
 imaginary
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Standard Equations of Hyperbolas, and Complex Conjugates and Division
 imaginary number
 Appears in these related concepts: Complex Numbers and The ComplexNumber System
 infinite
 Appears in these related concepts: Arithmetic Sequences, Sequences of Statements, and Summing Terms in an Arithmetic Sequence
 integer
 Appears in these related concepts: Scientific Notation, Total Number of Subsets, and Finding a Specific Term
 irrational number
 Appears in these related concepts: Zeroes of Polynomial Functions with Real Coefficients, Rational Coefficients, and Natural Logarithms
 operation
 Appears in these related concepts: Designing the Operation, Converting between Exponential and Logarithmic Equations, and Job security and people
 parabola
 Appears in these related concepts: Standard Form and Completing the Square, Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, and The Quadratic Formula
 point
 Appears in these related concepts: Circles, Relative Minima and Maxima, and Introduction: Polynomial and Rational Functions and Models
 rational number
 Appears in these related concepts: Domain of a Rational Expression, Rational Exponents, and Real Numbers: Basic Operations
 real number
 Appears in these related concepts: Graphing the Normal Distribution, Real Numbers, Functions, and Graphs, and Solving Problems with Inequalities
 real numbers
 Appears in these related concepts: Piecewise Functions, Linear Inequalities, and Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares
 relation
 Appears in these related concepts: Symmetry, Visualizing Domain and Range, and Functions and Their Notation
 series
 Appears in these related concepts: Charging a Battery: EMFs in Series and Parallel, Finding the General Term, and APA: Series and Lists
 sign
 Appears in these related concepts: The Intermediate Value Theorem, The Rule of Signs, and Polynomial Inequalities
 square
 Appears in these related concepts: Special Factorizations and Binomials, Radical Equations, and Matrix Multiplication
 whole number
 Appears in these related concepts: Basics of Graphing Polynomial Functions, Changing Logarithmic Bases, and Fractions
 zero
 Appears in these related concepts: Zeroes of Linear Functions, Rational Inequalities, and The Discriminant
Sources
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Cite This Source
Source: Boundless. “Radical Functions.” Boundless Algebra. Boundless, 18 Mar. 2016. Retrieved 01 May. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/radicalnotationandexponents14/radicalfunctions875517/