Download M098 Carson Elementary and Intermediate Algebra 3e Section 10.1 Objectives

M098
Carson Elementary and Intermediate Algebra 3e
Section 10.1
Objectives
1.
2.
3.
4.
Find the nth root of a number.
Approximate roots using a calculator
Evaluate radical functions
Find the domain of radical functions
Vocabulary
?nth root
Radicand
Index
n
The number b is an nth root of a number a if b = a.
The number under the radical symbol.
The number that indicates which root is to be used.
Prior Knowledge
Evaluating roots
Function – A relation in which every value in the domain (x) is paired with exactly one value in the range (y).
Domain – the set of all x values in a relation.
Function notation f(x). f is the name of the function and x is the variable.
Evaluating functions.
f(x) = 3x – 2
f(4) = 3(4) – 2 = 12 – 2 = 10
Solving linear inequalities. Remember to switch the direction of the inequality when both sides are multiplied
or divided by a negative number.
Interval notation and set builder notation.
x≥5
[5, ∞) – Interval notation. Smallest possible number is written on the left.
{ x | x ≥ 5 } – set builder notation
New Concepts
Real numbers are divided into rational and irrational numbers. So far we have dealt exclusively with rational
(fraction) numbers. This chapter adds the irrational component. Radicals that can not be simplified to
rational numbers are irrational.
radical expression:
index
radicand
1. Evaluating nth roots.
 We CANNOT take an even root of a negative number.
 4 or
4
 16 .
 We CAN take an odd root of a negative number. 3  8  2 .
 If the index is even, there are 2 roots: one is positive, one is negative.
The positive number is the principle root.
The radical symbol represents the positive root.
V. Zabrocki 2011
16  4 .
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M098
Carson Elementary and Intermediate Algebra 3e
Section 10.1
Example 1:
a.
49  7
e. 3  27  3
b.
 25 = not a real number
f.  3  125  5
c.  100  10
g. 5 32  2
d.  100  10
64
4
h. 3 
 
125
5
2. Evaluate radical functions.
Radical functions are functions that have a polynomial as the radicand. They are evaluated the same as
the linear functions we did earlier. Substitute the given value for the variable in the function.
f x  
Example 2:
f 3 
33  2 
3x  2
11
f  5 
3 5  2 
 13 Not a real number
3. Find the domain of radical functions.
The even root of a negative number does not exist so we must be careful to use only numbers that make
the radicand be 0 or positive. In Example 2, what numbers can be put in for x, so that the result will be a
real number?
Since the index is even, so the radicand (3x + 2) must be greater than or equal to 0.
3x + 2 ≥ 0
Solve this linear inequality.
3x ≥ -2
2
3
The domain is all real numbers greater than or equal to -2/3. As long as the input value for x is greater
than or equal to -2/3, the function will produce a real number value.
x  
2

 2

Domain:  ,   or x | x   
3

 3

Example 3 :
-2x + 6 ≥ 0
-2x ≥ -6
x≤3
V. Zabrocki 2011
What is the domain for f x  
 2x  6 ?
Since this is an even root, the radicand can not be
negative.
Divide both sides by -2. Remember to switch the
direction of the inequality.
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