Unit+7

=Unit 7 Radical Functions and Rational Exponents=

Vocabulary

 * composite function (p. 393) || like radicals (p. 374) || radical function (p. 409) ||
 * index (p. 364) || //n//th root (p. 363) || radicand (p. 364) ||
 * inverse functions (p. 403) || principal root (p. 364) || rational exponent (p. 379) ||
 * inverse relation (p. 401) || radical equation (p. 385) || rationalize the denominator (p. 370) ||

7-1 and 7-2 Objectives
For any real numbers //a// and //b//, and any positive integer //n//, if //a////n// = //b//, then //a// is an //n//th root of //b//. The **principal root** of a number with two real roots is the positive root. The principal //n//th root of //b// is written as. //b// is the **radicand** and //n// is the **index** of the radical. For any negative real number //a//, = | //a// | when //n// is even. If and  are real numbers, then • = , and, if //b// ≠ 0 , then  =. To **rationalize the denominator** of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical
 * 1) To simplify //n//th roots (p. 363)
 * 2) To multiply radical expressions (p. 368)
 * 3) To divide radical expressions (p. 369)



7-3 and 7-4 Objectives
Use FOIL to multiply binomial radical expressions. Binomials such as //a// + //b// and //a// − //b// are called conjugate expressions. If the denominator of a fraction is a binomial radical expression, multiply both the numerator and denominator of the fraction by the conjugate of the denominator to rationalize the denominator. The definition of **rational exponents** states that if the //n//th root of //a// is a real number and //m// is an integer, then =  and  =  =. If //m// is negative, //a// ≠ 0. The usual properties of exponents hold for rational exponents.
 * 1) To add and subtract radical expressions (p. 374)
 * 2) To multiply and divide binomial radical expressions (p. 375)
 * 3) To simplify expressions with rational exponents (p. 379)
 * Like radicals** have the same index and the same radicand. Use the distributive property to add or subtract them. Simplify radicals to find all the like radicals.



7-5 Objectives
To solve a **radical equation,** isolate a radical or a term with a rational exponent on one side of the equation. Then raise both sides of the equation to the same power. Check all possible solutions in the original equation to eliminate extraneous solutions.
 * 1) To solve radical equations (p. 385)

7-6 and 7-7 Objectives
The following are definitions of function operations. The composition of function //g// with function //f// is written as //g// //f// and is defined as (//g//  //f//)(//x//) = //g//(//f//(//x//)). The domain of the **composite function** //g// //f// consists of the values //a// in the domain of //f// such that //f//(//a//) is in the domain of //g//. If (//a//, //b//) is an ordered pair of a relation, then (//b//, //a//) is an ordered pair of its **inverse relation.** If a relation or function is described by an equation in //x// and //y//, you can interchange //x// and //y// to get the inverse. The inverse of a function is denoted by //f// − 1. If //f// and //f// − 1 are both functions, they are called **inverse functions,** and (//f// − 1 //f//)(//x//) = //x// and (//f//  //f// − 1)(//x//) = //x//.
 * 1) To add, subtract, multiply, and divide functions (p. 392)
 * 2) To find the composite of two functions (p. 393)
 * 3) To find the inverse of a relation or function (p. 400)

7-8 Objectives
A **radical function** such as //y// = has a restricted domain. For //y// =, the domain is the set of nonnegative real numbers. The graph of //y// = //a// + //k// is a translation //h// units horizontally and //k// units vertically of //y// = //a//.
 * 1) To graph radical functions (p. 408)