Module 1: Intro. to Radical Expressions and Functions

Transcription

1 Haberman MTH 95 Section IV: Radical Epressions, Equations, and Functions Module 1: Intro. to Radical Epressions and Functions The term radical is a fancy maematical term for e ings like square roots and cube roots at you may have studied in previous maematics courses. SQUARE ROOTS DEFINITION: A square root of a number a is a number c satisfying e equation c EXAMPLE: A square root of 9 is since 9. Anoer square root of 9 is since ( ) 9. DEFINITION: The principal square root of a number a is e nonnegative realnumber square root of RADICAL NOTATION: The principal square root of a is denoted by a. The symbol is called a radical sign. The epression under e radical sign is called e radicand. EXAMPLE: The square roots of 100 are 10 and 10. The principal square root of 100 is 10, which can be epressed in radical notation by e equation EXAMPLE: Are ere any real-number square roots of 5? According to e definition of square root (above) a square root of 5 would need to be a solution to e equation c 5. But ere is no real number which when squared is negative! Thus, ere is no real-number solution to is equation, so ere are no real numbers at are square roots of 5. In fact ere are no real-number square roots of ANY negative number!

2 Important Facts About Square Roots 1. Every positive real number has eactly TWO real-number square roots. (The two square roots of a are a and a.). Zero has only ONE square root: itself: NO negative real number has a real-number square root. EXAMPLE: Simplify e following epressions: m d m m m Here we need to use e absolute value since m could represent a negative number, but once m is squared and en square rooted, e result will be positive. d ( + ) + Again, we need e absolute value since + could represent a negative number.

3 The principal square root can be used to define e square root function: f ( ). Since negative numbers don t have square roots, e domain of e square root function is e set of non-negative real numbers: [ 0, ). Let s look at a graph of e square root function. We ll use a table-of-values to obtain ordered pairs to plot on our graph: f ( ) (, f( )) 0 0 (0, 0) 1 1 (1, 1) (, ) 9 (9, ) 16 (16, ) The graph of f ( ). Notice at e range of e square root function is e set of non-negative real numbers: [ 0, ). CUBE ROOTS DEFINITION: The cube root of a number a is a number c satisfying e equation c EXAMPLE: The cube root of 8 is since 8. Note at is e only cube root of 8. RADICAL NOTATION: The cube root of a is denoted by a.

4 EXAMPLE: The cube root of 1000 is 10 since could write Using radical notation, we The cube root of 1000 is 10 since ( 10) Using radical notation, we could write Important Fact About Cube Roots Every real number has eactly ONE real-number cube root. EXAMPLE: Simplify e following epressions k d. 8k 51 ( 8) k (k) k

5 5 d. 8 k ( k) k The cube root can be used to define e cube root function: g ( ). Since all real numbers have a real-number cube root, e domain of e cube root function is e set of real numbers, R. Let s look at a graph of e cube root function. g ( ) (, g ( )) 8 ( 8, ) 1 1 ( 1, 1) 0 0 ( 0, 0 ) 1 1 ( 1, 1 ) 8 ( 8, ) 7 ( 7, ) The graph of g ( ). Notice at e range of e cube root function is e set of real numbers, R. We can make a variety of functions using e square and cube roots.

6 6 EXAMPLE: Let w ( ) 5. Evaluate w(). Evaluate w(15). Evaluate w(0). d. What is e domain of w? w () () w ( 15) ( 15) w ( 0) ( 0) Since ere is no real number at is e square root of 5, we say at w(0) does not eist. d. Since only non-negative numbers have real-number square roots, we can only input into e function w -values at make e epression under e square root sign nonnegative, i.e., -values at make Thus, e domain of w is e set of real numbers greater an or equal to 5. In interval notation, e domain of w is 5, ).

7 7 EXAMPLE: Let ht () t+ 7. Evaluate h(t) if t 7. Evaluate h(t) if t 15. Evaluate h(t) if t 0. d. What is e domain of h? h + ( 7) h( 15) h + ( 0) d. Since every real number has a cube root, ere are no restrictions on which t-values at can be input into e function h. Therefore, e domain of h is e set of real numbers, R.

8 8 OTHER ROOTS We can etend e concept of square and cube roots and define roots based on any positive integer n. DEFINITION: For any integer n, an satisfying e equation n c n root of a number a is a number c RADICAL NOTATION: The principal n root of a is denoted by n a. EXAMPLE: What is e real-number root of 81? SOLUTION: Since 81 and ( ) 81 bo and are roots of 81. The principal root of 81 is (since principal roots are positive). We can write 81. EXAMPLE: What is e real-number 5 root of? 5 SOLUTION: Since e only 5 root of is. The principal 5 root of is (since is e only 5 root of ). We can write 5. The two eamples above epose a fundamental difference between odd and even roots. We only found one real number 5 root of, and 5 is an odd number, but we found two real number roots of 81 and is an even number. Important Facts About Odd and Even Roots 1. Every real number has eactly ONE real-number n root if n is odd.. Every positive real number has TWO real-number n roots if n is even. NOTE: Negative numbers do not have real-number even roots. So if n is even, we say at e n root of a negative number does not eist.

9 9 EXAMPLE: Simplify e following epressions t 6 6 Here we need to use e absolute value since could represent a negative number but once it is raised to an even power, e result will be positive t t Here we do not need to use e absolute value since if t is negative, once it is raised to an odd power e result will still be negative, and ere is a realnumber root of a negative number. 11 We can use n roots to define functions. p ( ) 1. EXAMPLE: Let Evaluate p(17). Evaluate p( 15). Evaluate p(). d. What is e domain of p? p ( 17)

10 10 p ( 15) Since ere is no real-number undefined. root of a negative number, we say at p( 15) is p () d. Since only non-negative numbers have real-number roots, we can only input into e function -values at make e epression under e radical sign non-negative, i.e., -values at make 1 0. Thus, e domain of p is [, )