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

radical
A root (of a number or quantity).

root
the number which,when plugged into the equation, will produce a zero.
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
Interactive Graph: Square Root of x
The graph of the function
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
Key Term Reference
 Parabola
 Appears in these related concepts: Nonlinear Systems of Inequalities, Types of Conic Sections, and Nonlinear Systems of Equations and ProblemSolving
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 cube root
 Appears in this related concept: Simplifying Radical Expressions
 e
 Appears in these related concepts: The Number e, Natural Logarithms, and Business Stakeholders: Internal and External
 exponent
 Appears in these related concepts: Logarithms of Products, Logarithms of Quotients, and Logarithms of Powers
 exponentiation
 Appears in these related concepts: Introduction to Exponents, Logarithmic Functions, and Simplifying Expressions of the Form log_a a^x and a(log_a x)
 expression
 Appears in these related concepts: Graphs of Equations as Graphs of Solutions, Simplifying, Multiplying, and Dividing Rational Expressions, and Bacterial Transformation
 function
 Appears in these related concepts: The Vertical Line Test, Visualizing Domain and Range, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphing Equations, Graphical Representations of Functions, and Reading Points on a Graph
 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: Domains of Rational and Radical Functions, Imaginary Numbers, and Sets of Numbers
 infinite
 Appears in these related concepts: Arithmetic Sequences, Summing Terms in an Arithmetic Sequence, and Sequences of Mathematical Statements
 integer
 Appears in these related concepts: Binomial Expansions and Pascal's Triangle, Scientific Notation, and Finding a Specific Term
 irrational number
 Appears in this related concept: Zeroes of Polynomial Functions With Rational Coefficients
 parabola
 Appears in these related concepts: The Quadratic Formula, What is a Quadratic Function?, and Completing the Square
 point
 Appears in these related concepts: Polynomial and Rational Functions as Models, Relative Minima and Maxima, and Graphing Quadratic Equations In Standard Form
 radical expression
 Appears in these related concepts: Introduction to Radicals, Adding, Subtracting, and Multiplying Radical Expressions, and Fractions Involving Radicals
 rational number
 Appears in these related concepts: Introduction to Rational Functions, Rational Exponents, and Basic Operations
 real number
 Appears in these related concepts: Introduction to Complex Numbers, Solving Problems with Inequalities, and Zeros of Polynomial Functions with Real Coefficients
 real numbers
 Appears in these related concepts: Linear Inequalities, Piecewise Functions, and Introduction to Domain and Range
 relation
 Appears in these related concepts: Symmetry of Functions, Functions and Their Notation, and What is a Linear Function?
 sign
 Appears in these related concepts: The Rule of Signs, Polynomial Inequalities, and The Intermediate Value Theorem
 square
 Appears in these related concepts: Matrix Multiplication, Radical Equations, and Factoring a Difference of Squares
 square root
 Appears in these related concepts: Graphs of Logarithmic Functions, The Discriminant, and Other Equations in Quadratic Form
 zero
 Appears in these related concepts: Rational Inequalities, Historical Traditions of Numerical Systems, and Zeroes of Linear Functions
 zeros
 Appears in these related concepts: Applications of the Parabola, Parts of a Parabola, and A Graphical Interpretation of Quadratic Solutions
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