Download Roots and Radical Expressions

Roots and Radical
Expressions
6-1
Vocabulary
Review
1. Circle the exponents in each expression.
217
23t 2
4a7b12
23x5
4y0
2. Underline the expression that is read five to the second power.
25
532
52
235
3. Parts of each expression below are blue. Write B if a base is blue, E if an exponent
is blue, or N if neither a base nor an exponent is blue.
E
24x54 1 6t
N
B
2r2 2 5
s5 1 7w2
Vocabulary Builder
Related Words: square root, cube root, nth root, power,
radical, index, radicand
index
n
6a
radicand
Definition: The nth root of a given number is a specific
number that, when it is used as a factor n times, equals the given number.
3
Using Symbols: 2 3 2 3 2 5 8, so !8
5 2.
Use Your Vocabulary
4. Draw a line from the number or symbol in Column A to each term in
4
Column B that best describes a part of !81
5 3.
Column A
Column B
4
radical
81
root
! index
3
radicand
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root
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radical
root (noun) root
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Key Concept The n th Root
If an 5 b, with a and b real numbers and n a positive integer, then a is an nth root of b.
If n is odd N
there is one real nth root of b, denoted
n
in radical form as !b.
If n is even N
• and b is positive, there are two real nth roots of b.
The positive root is the principal root (or principal
n
nth root). Its symbol is !b.
The negative root is its
n
opposite, or 2!b.
• and b is negative, there are no real nth roots of b.
The only nth root of 0 is 0.
5. Underline the correct words to complete the sentence.
4
Since the index of !81
is even / odd and the radicand is negative / positive , there
are two real fourth roots / no real fourth roots of 81.
Problem 1 Finding All Real Roots
Got It? What are the real fifth roots of 0, 21, and 32?
6. Cross out the question that will NOT help you find the roots.
What number is the index?
Is the radicand negative or positive?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
What number times 5 equals 21?
How many real roots of 0 are there?
7. Complete the equation to find the fifth root of 0.
0
303 0
303 0
50
8. The fifth root of 0 is 0 .
9. Complete the equation to find the fifth root of 21.
21 3 21 3 21 3 21 3 21 5 21
10. The fifth root of 21 is 21 .
11. Complete the equation to find the fifth root of 32.
2
3
2 3
2
3
2 3
2
5 32
12. The fifth root of 32 is 2 .
Problem 2 Finding Roots
Got It? What is each real-number root?
3
!227
4
!281
"(27)2
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!249
Lesson 6-1
4/9/09 10:01:40 AM
13. Circle the true equation.
(22)3 5 27
(23)3 5 27
(23)3 5 227
Underline the correct words to complete each sentence.
4
14. The index of !281
is even / odd and the radicand is positive / negative , so there
is one real root / are no real roots .
15. The index of !249 is even / odd , and the radicand is positive / negative ,
so there is one real root / are no real roots .
16. What is the principal square root of the square of a number?
Answers may vary. Sample: The principal square root of the square of a
_______________________________________________________________________
number is the absolute value of the original number.
_______________________________________________________________________
17. Simplify each root.
3
!227
4
!281
23
"(27)2
no real roots
!249
7
no real roots
Key Concept n th Roots of n th Powers
a if n is odd
.
u a u if n is even
3
3
18. The index of "
(22)3 is even / odd , so "
(22)3 5 22.
Problem 3
Simplifying Radical Expressions
Got It? What is a simpler form of the radical expression "81x 4 ?
19. Complete the equation to simplify the radical expression.
"81x 4 5
"92 Q
x2
R
2
5 9 x2
Problem 4
Using a Radical Expression
Got It? Some teachers adjust test scores when a test is difficult. One teacher’s
formula for adjusting scores is A 5 10!R, where A is the adjusted score and R is
the raw score. What are the adjusted scores for raw scores of 0 and 100?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
n
For any real number a, "an 5 e
20. Circle the first thing you should do to solve this problem.
Evaluate A 5 10!R for R 5 0 and R 5 100.
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Solve A 5 10!R for R.
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Underline the correct number to complete each sentence.
21. The lowest possible raw score is 0 / 10 / 15 / 100 .
The lowest possible adjusted score is 0 / 10 / 15 / 100 .
22. The highest possible raw score is 0 / 10 / 15 / 100 .
The highest possible adjusted score is 0 / 10 / 15 / 100 .
23. So, the curved scores for raw scores of 0 and 100 are 0 and 100 .
Lesson Check • Do you UNDERSTAND?
Error Analysis A student said the only fourth root of 16 is 2. Describe and correct
his error.
24. Multiple Choice If a is a fourth root of 16, which statement is true?
a ? a ? a ? a 5 16
a 5 164
4a 5 16
16
a5 4
25. Look at the statement you circled in Exercise 24. Circle the values of a below that
make the statement true.
264
24
22
2
4
64
26. There are 1 / 2 / 3 / 4 values of a that satisfy the statement you circled in Exercise 24.
Therefore, the number of fourth roots of 16 is 1 / 2 / 3 / 4 .
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
27. Describe the error the student made.
Answers may vary. Sample: The student may have forgotten that when
_______________________________________________________________________
n is even and b is positive, there are two nth roots of b, one positive
_______________________________________________________________________
and one negative.
_______________________________________________________________________
28. Complete the sentence to correct the error the student made.
The fourth root of 16 is 2 or 22 .
Math Success
Check off the vocabulary words that you understand.
nth root
principal root
radicand
index
Rate how well you can find nth roots.
Need to
review
0
2
4
6
8
Now I
get it!
10
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Lesson 6-1
4/9/09 10:02:01 AM
6-2
Multiplying and Dividing
Radical Expressions
PART 1
Vocabulary
Review
Write T for true or F for false.
F
1. The product of two numbers is always greater than either of the two factors.
F
2. When two negative numbers are multiplied together, the product is
a negative number.
T
3. The product of a number and its multiplicative inverse is 1.
T
4. The product of a fraction and 33 is a fraction equivalent to the original fraction.
Answers may vary. Samples: (22 )( 28 ), (0.3)(0.17), (4) Q 77 R
Vocabulary Builder
radical
index (noun)
IN
deks
index
n
Math Usage: In the expression !b, n is the index, and the
value of the expression is a number that, when multiplied
by itself n times, equals b.
radicand
5
is 5.
Example: The index of !18
Use Your Vocabulary
6. Circle the index in each expression.
4
!28
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3
!235
n
6a
7
!5.3
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5. Open-Ended Write three multiplication expressions: one where the product is
greater than either factor, one where the product is less than either factor, and one
where the product is equal to one factor.
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Property Combining Radicals: Products
n
n
n
n
n
If !a and !b are real numbers, then !a ? !b 5 !ab.
9. Draw a line from the product in Column A to an equivalent product in Column B.
Column A
Column B
4
3
3
Q!24 R
4
4
Q!4 R Q!9 R
Q!4 R Q!6 R
Q!3 R Q!8 R
3
3
4
4
4
Q!2 R Q!18 R
3
Q!24 R
Q!36 R
3
Q!36 R
Problem 1 Multiplying Radicals
4
4
Got It? Can you simplify the product of the radical expression !12
? !5?
Explain.
4
4
10. Circle the indexes in !12
? !5.
11. The indexes are the same / different .
4
!60
12. The simplified form of the product is
.
4
5
Got It? Can you simplify the product of the radical expression !7
? !7?
Explain.
4
5
13. Circle the indexes in !7 ? !7 .
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
14. The indexes are the same / different .
4
5
.
!7
? !7
15. The simplified form of the product is
5
5
Got It? Can you simplify the product of the radical expression !25
? !22?
Explain.
5
5
16. Circle the indexes in !25 ? !22 .
17. The indexes are the same / different .
5
18. The simplified form of the product is
!10
.
Problem 2 Simplifying a Radical Expression
3
3 3
Got It? What is the simplest form of "128x7 ? (Hint: Remember "
b 5 b .)
3
19. The index of "128x7 is 3 / 7 / 128 .
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Lesson 6-2, Part 1
4/9/09 10:02:58 AM
20. Circle the perfect cube factors.
3
3
"128x7 5 "43 ? 2 ? x 3 ? x 3 ? x
3
3
21. The simplest form of "128x7 is
4x 2!2x
.
Problem 3 Simplifying a Product
Got It? What is the simplest form of "15x 5y 3 ? "20xy 4 ?
22. Complete the equations to simplify "15x 5y 3 ? "20xy 4 .
"A15x 5y 3 B ? Q 20xy 4
5
"Q15 20 R ? Qx5y3
5
"Q3 ?
5
5
"Q3 ?
5
5 5
?
5
10x 3 y 3
R
xy 4 R
2
? 2
2
R ? Qx6 ? y7 R
2
? 2
2
R ? Q x2 ? x2 ? x2 ? y 2 ? y 2 ? y 2 ? y R
2 ? x 3 ? y 3 ? !3y
!3y
Lesson Check • Do you UNDERSTAND?
Reasoning For what values of x is "24x 3 real? Justify your reasoning.
23. Square roots of positive / negative numbers are real numbers.
24. The cube of a negative number is a positive / negative number.
25. The product of a negative number and a negative number is a
positive / negative number.
26. The expression "24x 3 is real when x is at least 0 / at most 0 . Explain
your reasoning.
Since the expression under the radical is the product of a negative
_______________________________________________________________________
number and the cube of another number, the cube must be a negative
_______________________________________________________________________
number (or 0) for the product to be nonnegative. The numbers whose
_______________________________________________________________________
cubes are negative (or 0) are the numbers less than or equal to 0.
_______________________________________________________________________
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
"15x 5y 3 ? "20xy 4 5
_______________________________________________________________________
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4/13/09 2:20:52 PM
6-2
Multiplying and Dividing
Radical Expressions
PART 2
Vocabulary
Review
Write T for true or F for false.
1. All mathematical expressions can be written as an equivalent expression with
a denominator of 1.
T
F
2. An expression can have a denominator equal to zero.
T
3. The expression above the fraction bar is the numerator.
4. Multiplying both the numerator and the denominator by the same non-zero
number results in an equivalent fraction.
T
Circle the numerator and underline the denominator in each expression.
5
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
5. 6
6.
25r 2
16
r2 1 s2
7. c 2 16
Vocabulary Builder
combine (verb) kum BYN
Main Idea: Combine means to put things together or to get a total.
Math Usage: To combine means to put together or add two like terms to get one term.
Example: The like terms 22x 3 and 7x 3 can be combined to get 5x 3 .
Use Your Vocabulary
8. Circle the expression that shows the like terms in 3x 2 1 1 1 4x 2 2 5 combined.
4x 2 2 1x 2
7x 2 2 4
3x 2 1 4x 2 1 1 2 5
Combine the like terms in each expression.
9. !3 1 2!3
3 !3
3
10. !5 1 !5 2 !5
3
!5
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4
11. !2 1 !4 1 2!4
4
!2 1 3 !4
Lesson 6-2, Part 2
4/9/09 10:03:32 AM
Property Combining Radicals: Quotients
n
n
If !a and !b are real numbers and b 2 0, then
n
!a
n
!b
n
5 Î ab .
9. Multiple Choice Choose the expression that is equivalent to
3
37
3
Q 2 !1 R Q x !1 R
3
!7
3
2
"4x
.
3 7
x Å4
Å4x 2
21x 2
Dividing Radical Expressions
Problem 4
Got It? What is the simplest form of
Q "50x 6 R
Q "2x 4 R
?
10. Complete the expression.
5
Q "2x4 R
5
50x 6
Å
2x 4
"
25x 2
5
5x
Use the Combining Radicals Property for Quotients.
Simplify under the radical sign.
Simplify the square root.
Rationalizing the Denominator
Problem 5
Got It? What is the simplest form of
3
!7x
3
"5y 2
?
11. The radicand in the denominator needs a 52 and a y to make 5y 2 a perfect cube.
12. You will need to multiply both the numerator and the denominator by the
3
expression Å
52 y
to rationalize the denominator.
13. Complete to show the rationalization of the denominator.
3
!7x
3
"5y 2
5
3
!7x
3
"5y 2
3
5
3
52 y
Å
175 x y
3 53
?
3
Multiply.
xy
Find the cube root of the denominator.
5?
Å
Rationalize the denominator.
y3
175
Å
3
5
52 y
Å
Å
5
?
3
Å
y
175x y
Simplify.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Q "50x6 R
5y
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14. Explain why both the numerator and the denominator are multiplied by the
expression used to rationalize the denominator.
Answers may vary. Sample: When a ratio is written with the same
_______________________________________________________________________
number in the numerator as in the denominator, its value is 1. When
_______________________________________________________________________
multiplying an expression by 1, in any form, the value of the expression
_______________________________________________________________________
is not changed.
_______________________________________________________________________
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
3 3y in simplest form.
Writing Explain how to write the expression Å
2
20xy
15. The variable y is in both the numerator and the denominator of the fraction.
16. Divide the numerator and denominator by the variable.
3
3
Å20xy
17. Rationalize the denominator.
3
Å20xy
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
3
5
5
5
5
3
!3
!20xy
3
3
!3
3 2
"
2 ? 5xy
?
3
"
2 ? 5 2x 2y 2
3
"
2 ? 5 2x 2y 2
3
"
3 ? 2 ? 5 2x 2y 2
3 3
"
2 ? 5 3x 3y 3
3
"
150x2y2
10xy
Math Success
Check off the vocabulary words that you understand.
simplest form of a radical
rationalize the denominator
Rate how well you can multiply and divide radical expressions.
Need to
review
0
2
4
6
8
Now I
get it!
10
341
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Lesson 6-2, Part 2
4/13/09 2:21:17 PM
Binomial Radical
Expressions
6-3
Vocabulary
Review
Circle the like terms in each group.
1.
3y 2
2y 2
2y
2.
b
bc
4bc
c
3.
5
18
5a
Vocabulary Builder
binomial (adjective) by NOH mee ul
Definition: A binomial expression is an expression made up of two terms.
Related Words: monomial, binomial expression, trinomial
Examples: monomial: a, x 2, 23, 17c 3, !5
trinomial: a 2 7 1 x, x 2 1 x 1 0.9, 23 2 ab 1 a, 17c 3 5 c 2 1 1,
b3 1 b 2 !5
Use Your Vocabulary
Write M if the expression is a monomial, B if the expression is a binomial, or T if
the expression is a trinomial.
T
4. 37 2 100x 3y 1 r
M
5. 257t 6
B
6. s 1 0.91r
T
7. 18a 2 1.4b7 1 3.85c14
Property Combining Radical Expressions: Sums and Differences
Use the Distributive Property to add or subtract like radicals.
n
n
n
a!x
1 b !x
5 (a 1 b) !x
Chapter 6
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n
n
n
a!x
2 b !x
5 (a 2 b) !x
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
binomial: a 2 7, x 2 1 0.9, 23 2 ab, 17c 3 1 1, b 2 !5
346
4/9/09 9:59:53 AM
8. Underline the words that make each sentence true.
To be like radicals, their indexes must be the same / different , and their radicands
must be the same / different .
To add or subtract two like radicals, you add or subtract their
radicands / coefficients .
Problem 1 Adding and Subtracting Radical Expressions
Got It? What is the simplified form of each expression?
3
7 !5 2 4!5
3x!xy 1 4x!xy
3
9. Are the radicals in 7 !5
2 4!5
like radicals?
10. Are the radicals in 3x!xy 1 4x!xy
like radicals?
Yes / No
Yes / No
3
11. Is 7 !5
2 4!5 simplified?
12. Is 3x!xy 1 4x!xy simplified?
Yes / No
Yes / No
13. Write the simplified form of each expression.
3
7 !5 2 4!5
3x!xy 1 4x!xy
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
3
7 !5
2 4 !5
7x !xy
Problem 2 Using Radical Expressions
Got It? In the stained-glass window design, the side of each small square
is 6 in. Find the perimeter of the window to the nearest tenth of an inch.
14. The length of the window is made up of the diagonals / sides of
three squares.
15. The width of the window is made up of the diagonals / sides of
two squares.
16. Multiple Choice Which is the length of the diagonal of a square with side s?
!2s
2s
s!3
s!2
17. Write the length of the diagonal of a square with side 6 in.
6 !2
18. Complete the following to find the length and the width of the window.
Length of the window:
/ 5 3 Q 6 !2 R 5
Width of the window:
18 !2
w 5 2 Q 6 !2 R 5
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12 !2
Lesson 6-3
4/9/09 10:00:19 AM
19. Complete the steps to find the perimeter of the window.
Perimeter 5 2/ 1 2w
5 2 Q 18 !2 R 1 2 Q 12 !2 R
Substitute for length and width.
5 36 !2 1 24 !2
Simplify.
5 60 !2
Add the coefficients of the like radicals.
5
Use a calculator to approximate to the nearest tenth.
Problem 3
84.9
Simplifying Before Adding or Subtracting
3
3
3
Got It? What is the simplified form of the expression !250
1 !54
2 !16?
20. Complete each factor tree to factor each radicand.
5
54
2
50
5
5
5
5
10
5
2
250 53 ∙ 2
16
2
27
2
3
2
3
9
3
3
54 2 ∙ 33
8
2
2
2
2
16 4
2
2
24
21. Complete the following by substituting the prime factorizations for 250, 54, and 16.
3
3
3
!250
1 !54
2 !16
5
3 53 3 2
"
1
3
"
2 ? 33 2
3
"
24
3
3
3
22. Circle the simplified form of the expression !250
1 !54
2 !16.
3
3
3
5!2
1 3!3
2 2!2
Problem 4
3
6!2
3
3
3!2
1 3!3
Multiplying Binomial Radical Expressions
Got It? What is the product A3 1 2!5B A2 1 4!5B ?
23. Circle the expression that shows the FOIL method.
A3 1 2!5B A2 1 4!5B
A3 1 2!5B A2 1 4!5B
3 ? 2 1 3 ? 4!5 1 2!5 ? 2 1 2!5 ? 4!5
24. The product A3 1 2!5B A2 1 4!5B 5
Chapter 6
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46 1 16 !5 .
3 ? 2 1 3 ? 4!5 1 2!5 ? 4!5
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
250
348
4/13/09 2:21:58 PM
Problem 5 Multiplying Conjugates
Got It? What is the product of the expression A6 2 !12B A6 1 !12B ?
25. Use the FOIL method to find the product.
A6 2 !12B A6 1 !12B 5 6 ? 6 1 6 ? !12 2 !12 ? 6 1 Q 2!12 R ? !12
5 36 2 12 5 24
Problem 6 Rationalizing the Denominator
Got It? How can you write the expression
denominator?
2 !7
with a rationalized
!3 2 !5
26. Circle the conjugate of the denominator. !5 2 !3 / !3 1 !5
2 !7
27. Use the conjugate of the denominator to write
with a
!3 2 !5
rational denominator.
2 !7
!3 2 !5
5
2 !7
!3 2 !5
?
!3 1 !5
!3 1 !5
5
2 !7A !3 1 !5B
A !3B 2 2 A !5B 2
5
2 !21 1 2 !35
325
5
2 !21 1 2 !35
22
5 2!21 2 !35
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Lesson Check • Do you UNDERSTAND?
Vocabulary Determine whether each of the following is a pair of like radicals. If so,
combine them.
3x!11 and 3x!10
2!3xy and 7!3xy
12!13y and 12!6y
28. Cross out the pairs that do NOT have the same index and the same radicand.
3x!11 and 3x!10
2!3xy and 7!3xy
9 !3xy
29. The sum of the like radicals is
12!13y and 12!6y
.
Math Success
Check off the vocabulary words that you understand.
like radicals
binomial radical expressions
Rate how well you can add and subtract radical expressions.
Need to
review
0
2
4
6
8
Now I
get it!
10
349
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Lesson 6-3
4/9/09 10:00:42 AM
6-4
Rational Exponents
PART 1
Vocabulary
Review
1. Use the terms listed in the blue box below to complete the Venn diagram. Then,
give an example for each. Examples may vary. Samples are given.
Irrational Numbers
Integers
Rational Numbers
Natural Numbers
Whole Numbers
Real Number Sets
Irrational
Numbers
Example:
Rational Numbers
Example: 3
4
Integers
Example: 3
C
Natural Numbers
Example: 0
Example: 1
Vocabulary Builder
convert (verb) kun VURT
Related Words: conversion (noun), convertible (adjective)
What It Means: To convert means to rewrite or to change to another form.
Use Your Vocabulary
2. Circle the expression that shows 34 converted to a decimal.
1.3
Chapter 6
HSM11A2LC_C06_354-359.indd 354
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Whole Numbers
0.34
0.75
354
4/9/09 9:56:25 AM
Complete each sentence with the correct form of the word convert.
3. NOUN Travelers to Europe calculate the 9 of dollars to euros.
conversion
4. ADJECTIVE A 9 sofa is a couch by day and a bed by night.
convertible
convert
5. VERB To 9 fractions to decimals, divide the numerator by the denominator.
Problem 1 Simplifying Expressions With Rational Exponents
1
Got It? What is the simplified form of 642 ?
1
1
n
6. Circle the expression that shows 642 rewritten as a radical. (Hint: x n 5 !x)
1
!64
1
2 !64
2
!64
1
7. The simplified form of 64 2 is 8 .
Key Concept Rational Exponent
If the nth root of a is a real number and m is an integer, then
m
1
n
n
m
n
an 5 !a
and a n 5 "am 5 (!a ) .
If m is negative, then a 2 0.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
8. Draw a line from the radical expression in Column A to its rational exponent form
in Column B.
Column A
Column B
3
4!x
4x 3
4 3
"
x
x4
1
3
Problem 2 Converting Between Exponential and Radical Form
Got It? What are w28 and w 0.2 in radical form?
5
9. Convert w 0.2 to w raised to a power that is a fraction in lowest terms.
1
w5
m
n
10. Identify the values of a, m, and n for each expression using the rule a n 5 "am .
w28
5
a5 w
w 0.2
m 5 25
n5 8
a5 w
m5
1
n5
5
11. Use your values for a, m and n to write each expression in radical form.
w28
5
w 0.2
5
!w
8 25
"
w
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Lesson 6-4, Part 1
4/13/09 2:22:39 PM
Using Rational Exponents
Problem 3
Got It? Planetary Motion Kepler’s third law of Orbital Motion states that the
period P (in Earth years) it takes a planet to complete one orbit of the sun is
3
approximately P 5 d 2 where d is the distance (in astronomical units, AU) from the
planet to the sun. Find the approximate length (in Earth years) of a Venusian year if
Venus is 0.72 AU from the sun.
12. Complete the problem-solving model below.
Know
Need
Plan
the distance from Venus
to the sun, and the
formula for the length
of a planet year
the length of a
Venusian year
Use the formula to find
the period.
13. Use the formula to find the length of a Venusian year.
3
P 5 d2
3
P 5 (0.72)2
P N 0.611 years
Lesson Check • Do you UNDERSTAND?
1
1
1
1
14. Circle the expressions that simplify to real numbers.
3
3
!264
2!64
!264
!64
15. Simplify each expression, if possible.
3
3
!264
2!64
!264
!64
24
24
can’t
simplify
8
16. Explain why the statements are not both true.
Answers may vary. Sample: There is no real square root of a negative
_______________________________________________________________________
1
1
number, so (264)2 is undefined. 264 2 is the opposite of the square root
_______________________________________________________________________
of 64, or 28. The cube root of a negative number exists, equals the
_______________________________________________________________________
opposite of the cube root of the absolute value of the number.
_______________________________________________________________________
_______________________________________________________________________
Chapter 6
HSM11A2LC_C06_354-359.indd 356
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Reasoning Explain why (264)3 5 2643 but (264)2 u 2642 .
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6-4
Rational Exponents
PART 2
Vocabulary
Review
Circle the exponent(s) in each expression.
3. x23 1 4z 2
2. 23.5 2 y 5
1. 284
Vocabulary Builder
rewrite (verb) ree RYT
Definition: To rewrite means to write in a different form or matter.
Math Usage: Rewrite a number or expression in a simpler or more useful form.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Example: To rewrite a decimal as a percent, move the decimal point two places to
the right.
Use Your Vocabulary
4. Circle the expression that shows 45 rewritten as a decimal.
1.25
4q5.00
0.80
5q4.00
0.45
Key Concept Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator equals 0.
am ? an 5 am1n
(am)n 5 amn
(ab)m 5 ambm
a2m 5 1m
am
5 am2n
an
a
a
Q b R 5 bm
a
m
m
Complete each example below.
4
5. g 3 ? g 4 5 g 31
1
8. 226 5
2
6
5
5g
1
64
7
4
6. (h3)4 5 h3?
9.
3
d8
5 d 82
3
d
5d
12
5
7. (5y)2 5 5
2
32
3 2
10. Q 4 R 5
4
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5h
2
?y
5
2
9
16
Lesson 6-4, Part 2
4/9/09 9:57:16 AM
Combining Radicals With Like Radicands
Problem 4
Got It? What is
11. Rewrite
"x 3
3 2 in simplest form?
"
x
"x 3
3 2 in exponential form.
"
x
3
"x 3
3 2 5
"
x
x2
2
x3
12. The bases of the factors are the same / different, so the property you should use to
simplify the exponential form is
am
a m
am
5 a m2n / Q b R 5 m .
b
an
13. To subtract fractions, rewrite them with the same numerator / denominator .
14. Write
"x 3
3 2 in simplest form.
"
x
5
x6
4 B
Got It? What is !3 A!3
in simplest form?
4
15. Rewrite !3 Q !3 R in exponential form.
4
!3 A !3 B 5 32 ? 34
1
1
simplify the exponential form is (ab)m 5 a mb m / a m ? a n 5 a m1n .
4
17. Write !3 Q !3 R in simplest form.
4
3
34 or !27
Simplifying Numbers With Rational Exponents
Problem 5
Got It? What is 3225 in simplest form?
3
18. Solve using two different methods. Complete each method.
Method 1
3225 5 (2
3
Method 2
5
)25
3225 5
3
3
1
3
32 5
5 (2)
5 (2)
5 ?235
23
5
1
5
1
8
5
5
Q"
32 R
3
1
5
Q 2 R
Chapter 6
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1
8
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
16. The bases of the factors are the same / different so the property you should use to
3
358
4/13/09 2:23:03 PM
Problem 6 Writing Expressions in Simpler Form
1
Got It? What is (8x15)23 in a simpler form?
19. The expression is a product / quotient raised to a power so the property you can
a m
am
use to simplify the exponential form is (ab)m 5 a mb m / Q b R 5 m .
b
20. Simplify the exponential form.
(8x15)23 5 8
213
21
? x15? 3
58
213
?x
1
1
5
8
5
25
1
?
1
3
x
5
1
2x 5
Lesson Check • Do you UNDERSTAND?
Error Analysis Explain why this simplification is incorrect.
1
21. The radical form of 52 is
!5
1
5(4) – 5(52)
1
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
The radical form of 5(5 ) is 5 !5 .
1
2
The radical form of 25 is
1
5(4 – 5 2)
.
1
20 – 25 2
15
!25 .
22. Explain why the simplification shown is incorrect.
1
Answers may vary. Sample: The student incorrectly simplified 5(52).
_______________________________________________________________________
The answer should be 20 2 5 !5.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
rational exponent
exponential form
radical form
Rate how well you can simplify expressions with rational exponents.
Need to
review
0
2
4
6
8
Now I
get it!
10
359
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Lesson 6-4, Part 2
4/13/09 2:23:08 PM
6-5
Solving Square Root and
Other Radical Equations
PART 1
Vocabulary
Review
1. Circle the equation that is equivalent to !x 1 1 5 5.
!x 1 1 5 10
!x 1 2 5 6
!x 1 1 1 3 5 8
2. Draw a line from each equation in Column A to its solution in Column B.
Column A
Column B
31x55
210
y 2 4 5 26
10
2z 5 20
2
w
5 5 22
22
Circle the radicand in each expression.
4.
7 2 !wz 1 2
5.
!y 1 7
Vocabulary Builder
reciprocals
reciprocal (noun) rih SIP ruh kul
a and b
b
a
where a 0 and b 0
Related Term: multiplicative inverse
Definition: Two numbers are reciprocals if their product is 1..
Main Idea: To write the reciprocal of a fraction, switch the numerator
and denominator.
7
Examples: 3 and 13 , 247 and 24 , 16 and 6
Use Your Vocabulary
Write T for true or F for false.
T
6. The reciprocal of 21 is itself.
F
7. A decimal has no reciprocal.
T
8. The reciprocal of a negative
number is negative.
T
9. The only real number without a
reciprocal is 0.
Chapter 6
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Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
!2x 2 3
3.
364
4/9/09 10:08:14 AM
Problem 1 Solving a Square Root Equation
Got It? What is the solution of !4x 1 1 2 5 5 0?
10. Circle the first step in solving the equation.
Isolate the square root.
Square each side.
11. Underline the correct reason for each step.
!4x 1 1 5 5
Isolate the square root / variable .
4x 1 1 5 25
Take the square root / square of each side to remove the radical.
4x 5 24
Subtract 1 to isolate the radical / variable term.
x56
Divide / Multiply by 4 to solve for x.
Problem 2 Solving Other Radical Equations
2
Got It? What is the solution of 2(x 1 3)3 5 8?
12. Follow the steps to find the solution.
1
Isolate the radical term.
2
2(x 3) 3 8
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
(x 3) 3 4
2
Raise each side of the equation to the reciprocal power.
+(x
3
2
3) 3
,
3
2
4
3
2
Solve for x.
Ux 3U 8
2
3
x 3 8
x 5 or x 11
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3
Since the numerator of 2 is even, +(x 3) 3 , 2 Ux 3U.
Lesson 6-5, Part 1
4/13/09 2:23:44 PM
Problem 3
Using Radical Equations
3
V
Got It? For Meteor Crater in Arizona, the formula d 5 2 ? Î 0.3
relates the diameter
d of the rim in meters to the volume V in cubic meters. Suppose the diameter of a
similarly shaped crater is 1 kilometer. What is the volume?
13. Which equation can you use to find the volume?
10 5 2 ?
3 V
100 5 2 ?
Å0.3 3
3 V
1000 5 2 ?
Å0.3
3 V
Å0.3
10,000 5 2 ?
3 V
Å0.3
14. To isolate the radicand, divide each side of the equation by 1 / 2 / 3 / 10 and raise
each side of the equation to the 1st / 2nd / 3rd / 10th power.
15. Solve the equation.
V
1000 5 2 ? 3Å0.3
V
500 5 3Å0.3
V
5003 5 0.3
V 5 0.3 ? 5003 5 37,500,000
16. The volume of the crater is
37,500,000
m3 .
Compare and Contrast How is solving a quadratic equation using square roots
similar to solving a square root equation? How is it different?
Write T for true or F for false.
T
17. The first step in solving a square root equation is to isolate the radical.
F
18. To solve a square root equation, take the square root of each side.
F
19. You can solve a square root equation using the Quadratic Formula.
20. Describe the similarities and differences between solving a quadratic equation and
solving a square root equation.
Answers may vary. Sample: They are similar since they involve the
_______________________________________________________________________
inverse operations of squaring and taking a square root. They are
_______________________________________________________________________
different because it is not necessary to isolate the quadratic term to
_______________________________________________________________________
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Lesson Check • Do you UNDERSTAND?
solve a quadratic equation.
_______________________________________________________________________
Chapter 6
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4/13/09 2:23:47 PM
6-5
PART 2
Solving Square Root and
Other Radical Equations
Vocabulary
Review
1. Draw a line from each square root expression in Column A to an equivalent
expression in Column B.
Column A
Column B
!100
10!x
!100x
10
"100x 2
10!10
!1000
10u x u
2. Circle the square root equation that is equivalent to !x 2 2 5 4.
!x 2 2 5 4
!x 2 1 5 5
!x 2 2 1 7 5 11
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
eliminate (noun) ee LIM uh nayt
Definition: To eliminate something is to remove or get rid of it.
Math Usage: To eliminate a term in an equation, use the inverse operation.
Main Idea: To isolate a variable term in an equation, use inverse operations, doing
the same thing to each side of the equation to keep it true.
Use Your Vocabulary
3. To eliminate the 3 from x 1 3 5 5, add 3 to / subtract 3 from each side.
4. To eliminate the 6.7 from t 2 6.7 5 5, add 6.7 to / subtract 6.7 from each side.
5. To eliminate the 4 from 4y 5 252, multiply / divide each side by 4.
6. To eliminate the radical in !z 2 1 5 10, square / take the square root of each side.
s
7. To eliminate the 17 in the equation 17
5 0.5, multiply / divide each side by 17.
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Lesson 6-5, Part 2
4/9/09 10:08:59 AM
Solving a square root equation may require that you square each side of the
equation. This can introduce extraneous solutions.
Problem 4
Checking for Extraneous Solutions
Got It? What is the solution of !5x 2 1 1 3 5 x? Check your results.
8. Use the justifications at the right to complete each step.
!5x 2 1 1 3 5 x
Write the original equation.
!5x 2 1 5 x 2 3
( !5x 2 1)2 5
(x 2 3)2
5x 2 1 5 x 2 1 26 x 1 9
Isolate the radical.
Square each side of the equation.
Simplify.
0 5 x 2 2 11 x 1 10
Combine like terms.
0 5 Qx 2
Factor.
x5
1 R Q x 2 10 R
1 or x 5 10
Use the Zero-Product Property.
9. Substitute each value into the original equation to check the solutions.
"5Q
4 1301
"5Q 10 R 2 1 1 3 0 10
" 49 1 3 0 10
2 1301
7
1 R21 1301
"
!5x 2 1 1 3 5 x
5 01
1 3 0 10
10 5 10
10. The solution 1 is extraneous.
11. Multiple Choice What can cause an extraneous solution?
raising each side of the equation to an odd power
raising each side of the equation to an even power
adding the same number to each side of an equation
dividing each side of an equation by the same number
12. When should you check for extraneous solutions? Explain. Answers may vary. Sample:
You should check for an extraneous solution any time you raise
_______________________________________________________________________
both sides of an equation to an even power.
_______________________________________________________________________
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
!5x 2 1 1 3 5 x
Problem 5 Solving an Equation With Two Radicals
Got It? What is the solution of !5x 1 4 2 !x 5 4?
Chapter 6
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368
4/13/09 2:24:04 PM
13. The equation has been solved below. Write the letter of the reason that justifies
each step. Use the reasons in the box.
A Addition Property of Equality
A
!5x 1 4 5 !x 1 4
B Square each side.
5x 1 4 5 ( !x 1 4)2
B
5x 1 4 5 x 1 8!x 1 16
E
D Factor.
4x 2 12 5 8!x
A
E Simplify.
x 2 3 5 2!x
C
(x 2 3)2 5 4x
B
x 2 2 6x 1 9 5 4x
E
x 2 2 10x 1 9 5 0
A
(x 2 9)(x 1 1) 5 0
D
C Division Property of Equality
F Zero-Product Property
F
x 5 9 or x 5 21
14. Only the solution x 5 21 > x 5 9 satisfies the original equation.
Lesson Check • Do you UNDERSTAND?
Vocabulary Which value, 12 or 3, is an extraneous solution of x 2 6 5 !3x?
Explain your reasoning.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
15. The solution x 5 12 satisfies / does not satisfy the original equation. The solution
x 5 3 satisfies / does not satisfy the original equation.
16. The solution x 5 12 > x 5 3 is an extraneous solution of x 2 6 5 !3x.
17. Explain why the solution is extraneous. Answers may vary. Sample:
The solution is extraneous because, while it is a solution of the related
_______________________________________________________________________
quadratic equation, it does not make the original equation true.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
radical equation
square root equation
Rate how well you can solve square root and other radical equations.
Need to
review
0
2
4
6
8
Now I
get it!
10
369
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Lesson 6-5, Part 2
4/9/09 10:09:30 AM
Function Operations
6-6
Vocabulary
Review
1. In function notation, gx / g(x) / x(g) is read g of x.
2. Circle the equation that shows a function rule.
x 1 17y 5 24.7
f (x) 5 14x 2 0.3
15z(13t)
3. The function rule f (t) 5 1.83t represents the cost of a number of tons of wheat t.
The number of tons of wheat is the input / output .
The output is the cost of the wheat / number of tons of wheat .
composite (adjective) kum PAHZ it
Related Words: composite function,
composite number
Domain of f
Range of f
Input
of f,
x
Output
of f,
f(x)
Main Idea: Something that is
composite is made up of more than
one thing.
Math Usage: A composite function
uses the output of one function as
the input of a second function.
Input
of g,
f(x)
Output
of g,
g(f(x))
Domain of g
Range of g
Use Your Vocabulary
Complete each sentence with the correct form of the word composite.
composite
composition
compose
4. VERB The musician worked to 9 a new piece of music.
5. ADJECTIVE A 9 number has more than two factors.
6. NOUN The poster was a 9 of photos and famous quotes.
Chapter 6
HSM11A2LC_C06_374-377.indd 374
compose
composite
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
composition
374
4/9/09 10:10:05 AM
Key Concept Function Operations
Addition
( f 1 g)(x) 5 f (x) 1 g(x)
Subtraction
( f 2 g)(x) 5 f (x) 2 g(x)
Multiplication
( f ? g)(x) 5 f (x) ? g(x)
Division
Q g R (x) 5 g(x) , g(x) 2 0
f
f (x)
The domains of the sum, difference, product, and quotient functions consist of the
x-values that are in the domains of both f and g. Also, the domain of the quotient
function does not contain any x-value for which g(x) 5 0.
7. Let f (x) 5 x 2 1 2 and g(x) 5 x 2 6. Complete the steps for finding f (x) 1 g(x).
f (x) 1 g(x) 5 (x 2 1 2) 1
5 x2 1 x 2
(x 2 6)
4
Problem 1 Adding and Subtracting Functions
Got It? Let f (x) 5 2x 2 1 8 and g(x) 5 x 2 3. What are f 1 g and f 2 g ? What
are their domains?
8. Circle f 1 g .
2x 2 1 x 1 8
2x 2 1 x 1 5
x2 1 5
x2 1 8
2x 2 2 x 1 5
x2 1 5
x 2 1 11
9. Circle f 2 g .
2x 2 2 x 1 11
10. Write the domain of f.
11. Write the domain of g.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
all real numbers
all real numbers
all real numbers
12. The domain of both f 1 g and f 2 g is
.
Problem 2 Multiplying and Dividing Functions
Got It? Let f (x) 5 3x 2 2 11x 2 4 and g(x) 5 3x 1 1. What are f ? g and
f
g
and
their domains?
13. Circle f ? g .
3x 2 2 11x 2 4 1 3x 1 1
A3x 2 2 11x 2 4B A3x 1 1B
3x 2 2 11x 2 4
3x 1 1
f
14. Circle g .
3x 1 1
3x 2 2 11x 2 4
A3x 2 2 11x 2 4B A3x 1 1B
15. Simplify the product.
3x 2 2 11x 2 4
3x 1 1
16. Simplify the quotient.
x24
9x3 2 30x 2 2 23x 2 4
17. For what values of x does g(x) 5 0?
213
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Lesson 6-6
4/9/09 10:10:22 AM
f
18. Write the domain of f ? g .
19. Write the domain of g .
all real numbers except x 5 213
all real numbers
Key Concept Composition of Functions
The composition of function g with function f is written g + f and is defined as
(g + f )(x) 5 g( f (x)). The domain of g + f consists of the x-values in the domain of f
for which f (x) is in the domain of g.
(g + f )(x) 5 g( f (x))
1. Evaluate f (x) first.
1
2. Then use f (x) as the input for g.
2
Function composition is not commutative, since f (g(x)) does not always equal g( f (x)).
If f (x) 5 x 2 and g(x) 5 x 1 3, find each composition.
20. g + f 5 g(x 2) 5
x2
13
21. f + g 5 f (x 1 3) 5 (x 1 3)
Problem 3
2
5 x2 1
x1
6
9
Composing Functions
Got It? Let f (x) 5 x 2 5 and g(x) 5 x 2 . What is ( f + g)(23)?
22. Find g(23).
9
23. Use g(23) as the input for f (x).
( f + g)(23) 5 f(g(23)) 5 f Q
9
R5 925 5
4
Method 2
24. Simplify ( f + g)(x) 5 f (g(x)).
f (g(x)) 5 f Q
5
x2
x2
R
25. Use the rule for f (g(x)) to find f (g(23)).
( f + g)(23) 5 (23)2 2 5 5
4
25
Problem 4 Using Composite Functions
Got It? A store is offering a 15% discount on all items. Also, employees get a 20%
employee discount. Write composite functions.
Model taking the 15% discount and then the 20% discount.
Model taking the 20% discount and then the 15% discount.
Chapter 6
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Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Method 1
376
4/9/09 10:10:37 AM
26. Let x be the price of an item. Write functions to model each discount.
C(x) 5 cost using the storewide discount 5 x 2
D(x) 5 cost using the employee discount 5 x 2
0.15 x 5
0.2
0.85 x
x5
0.8
x
27. Draw a line from each composition in Column A to its rule in Column B.
Column A
Column B
(D + C)(x)
0.85(0.8x)
(C + D)(x)
0.8(0.85x)
28. Simplify each function.
(D + C)(x) 5
(C + D)(x) 5
0.68x
0.68x
29. If you were an employee, which discount would you take first? Why?
Answers may vary. Sample: Either discount can be taken first, since in
_______________________________________________________________________
both cases the result is that you pay 68% of the original price.
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
Open-Ended Find two functions f and g such that f (g(x)) 5 x for all real numbers x.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
30. If g(x) 5 x 1 3, write a function f (x)
to give f (g(x)) 5 x.
31. If g(x) 5 2x, write a function f (x)
to give f (g(x)) 5 x.
f(x) 5 x2
f(x) 5 x 2 3
32. Write a function g(x). Then, find f (x) such that f (g(x)) 5 x.
Answers may vary. The functions should be inverses of one another.
Math Success
Check off the vocabulary words that you understand.
composite function
function operations
Rate how well you can find the composition of two functions.
Need to
review
0
2
4
6
8
Now I
get it!
10
377
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Lesson 6-6
4/13/09 2:24:56 PM
6-7
Inverse Relations and
Functions
PART 1
Vocabulary
Review
1. Underline the correct term to complete each sentence.
The domain of a relation is the set of inputs, also called the x- / y- coordinates of
the ordered pair.
The range of a relation is the set of outputs, also called the x- / y- coordinates of the
ordered pair.
2. Use the relation {(4, 5), (6, 7), (12, 20), (8, 3), (2, 7)}. Write the domain and the range
of the relation.
Domain
Range
2, 4, 6, 8, 12
3, 5, 7, 20
inverse (noun) in VURS
Related Words: opposite, reverse
Math Usage: The inverse of a function, is found by reversing the order of the
elements in the ordered pairs.
Example: The inverse of the function {(1, 2), (2, 4), (3, 6)} is {(2, 1), (4, 2), (6, 3)}.
Use Your Vocabulary
3. Complete the diagram below. Use the relation r {(0, 1), (2, 3), (4, 1), (8, 3)}.
Relation r
Inverse of Relation r
Domain
Range
Domain
Range
0, 2, 4, 8
1, 3
1, 3
0, 2, 4, 8
Chapter 6
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Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
382
4/13/09 2:25:23 PM
Problem 1 Finding the Inverse of a Relation
Got It? What are the graphs of t and its inverse?
4. Complete the table of values for the relation t.
Relation t
Inverse of Relation t
x
y
x
y
0
5
5
0
1
4
4
1
2
3
3
2
3
3
3
3
5. Plot the points in both the Relation t table and the Inverse of Relation t table.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Relation t
y
5
4
3
2
1
5 4 3 2 1 O
1
2
3
4
5
Inverse of Relation t
y
5
4
3
2
1
x
5 4 3 2 1 O
1
2
3
4
5
1 2 3 4 5
x
1 2 3 4 5
Problem 2 Finding an Equation for the Inverse
Got It? What is the inverse of y 5 2x 1 8?
6. Switch the x and y values in the function.
y 5 2x 1 8
function
inverse
x
52 y
18
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Lesson 6-7, Part 1
4/9/09 10:06:24 AM
7. Solve the inverse equation for y.
x 5 2y 1 8
x 2 8 5 2y
x28
2
Problem 3
5y
Graphing a Relation and Its Inverse
Got It? What are the graphs of y 5 2x 1 8 and its inverse?
8. Complete the table for y 5 2x 1 8.
y 5 2x 1 8
x
6
4
2
0
y
4
0
4
8
9. Complete the table for the inverse of y 5 2x 1 8.
x28
2
y
x
4
0
4
8
y
6
4
2
0
8
y 2x 8
4
10. Plot and draw a line through the points from the
y 5 2x 1 8 table.
8
4
11. On the same grid, plot and draw a line through
the points from the inverse of y 5 2x 1 8 table.
O
4
8
12. Draw a dashed line to show the line that reflects the
equation y 5 2x 1 8 to its inverse.
Lesson Check • Do you UNDERSTAND?
Reasoning A function consists of the pairs (2, 3), (x, 4), and (5, 6). What values,
if any, may x not assume?
13. Each x-value in the domain of a function corresponds to
exactly one y-value / many y-values in the range.
14. What values, if any, may x not assume?
Answers may vary. Sample: x cannot equal 2 or 5.
_______________________________________________________________________
y x
x
4
8
yx8
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
y5
_______________________________________________________________________
Chapter 6
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4/9/09 10:06:30 AM
6-7
Inverse Relations and
Functions
PART 2
Vocabulary
Review
Write T for true or F for false.
F
1. All relations are functions.
T
2. All relations have inverses.
Vocabulary Builder
one-to-one (adjective) wun too wun
Math Usage: A function is one-to-one if each y-value has only one x-value. Thus,
the inverse of a one-to-one function is also a function.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Example: All linear functions that are not constant functions are one-to-one.
Use Your Vocabulary
3. A function is one-to-one. Given the points {(22, 21), (2, 3), (4, 5.2), (7, 3.5)} in the
function, write the points in the inverse function.
{(21, 22), (3, 2), (5.2, 4), (3.5, 7)}
Problem 4 Finding an Inverse Function
Got It? Let g(x) 5 6 2 4x. What are the domain and the range of g? What is the
inverse of g? What are the domain and range of g21 ? Is g21 a function? Explain.
4. Use the justifications at the right to write g21 .
y
5 6 2 4x
Rewrite the equation using y.
x 5624 y
Switch x and y.
x2
6 5 24y
Subtract.
x2
6
24
5y
Divide.
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Lesson 6-7, Part 2
4/9/09 10:06:50 AM
214x 1 32
5. The inverse of g(x) is g 21(x) 5
.
6. For every real input value of g 21 , there is / is not an output value. The domain and
range of g are all real numbers / restricted .
7. For each y-value of g 21 , there is 1 / more than 1 x-value. So, g 21 is / is not
a function.
Problem 5
Finding the Inverse of a Formula Function
2
v
Got It? The function d 5 19.6
relates the distance d, in meters, that an object
has fallen to its velocity v, in meters per second. Find the inverse of this function.
What is the velocity of the cliff diver in meters per second as he enters the water?
8. Solve the function for v.
24 meters
2
v
d 5 19.6
19.6d 5 v 2 m !19.6d 5 v
9. Let d 5 24 meters. Write the value of the velocity v, to the nearest hundredth
meter per second, of the diver as he enters the water.
21.69
If f and f 21 are inverse functions, then Q f 21 + f R (x) 5 x and Q f + f 21 R (x) 5 x for x in
the domains of f and f 21, respectively.
Problem 6 Composition of Inverse Functions
Got It? Let g(x) 5 x
4
21
1 2 . What is g (x)?
10. Complete each step.
x 5 4
y 1 2
Rewrite the inverse function.
x Qy 1 2 R 5 4
( y 1 2) 5
y5
Multiply.
4
Divide.
x
4
x
2 2
11. The inverse of g(x) is g21(x) 5
Got It? Let g(x) 5 x
Chapter 6
HSM11A2LC_C06_382-387.indd 386
4
1 2 . What is
Subtract.
4
x
22
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Key Concept Composition of Inverse Functions
.
Ag + g21 B(0)? What is Ag21 + gB(0)?
386
4/13/09 2:25:58 PM
12. Complete each equation.
Ag + g21 B(0) 5 g
4
2
Ag21 + gB(0) 5 g21
2
0
4
0 1 2
13. Which is undefined, Ag + g21 B(0) or Ag21 + gB(0)? Explain.
Ag + g21 B(0). Explanations may vary. Sample: Division by zero is undefined.
_______________________________________________________________________
_______________________________________________________________________
14. Simplify the other expression.
4
4
21
Ag21 + gB(0) 5 g210 1
2 5 g (2) 5 2 2 2 5 0
Lesson Check • Do you know HOW?
1
For h(x) 5 (x 1
, find h21(x), h21(4), and the value(s) of x for which the equality
2)
Ah + h21 B(x) 5 x does not hold.
15. Complete each step to find h21(x).
16. Find h21(4).
1
x 5y1
2
h21(x) 5
x Qy 1
h21(4) 5
2 R51
1
x
1
22
22
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
4
Qy 1 2 R 5
y5
1
5 21.75
x
1
x
22
18. Is h21(x) defined for every real number x?
17. Is h(x) defined for every real number x?
Yes / No
Yes / No
19. Circle the values of x for which the equality Ah + h21 B(x) 5 x does not hold.
22
0
1
2
4
Math Success
Check off the vocabulary words that you understand.
inverse relation
inverse function
one-to-one function
Rate how well you can find the inverse of a relation or function.
Need to
review
0
2
4
6
8
Now I
get it!
10
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Lesson 6-7, Part 2
4/9/09 10:07:30 AM
6-8
Graphing Radical
Functions
Vocabulary
Review
Write T if the figures show a translation or N if they do NOT show a translation.
1.
T
2.
N
3.
N
Vocabulary Builder
VUR
tih kul
vertical
Related Word: horizontal
Main Idea: A vertical line is straight up and down.
A vertical shift moves something up or down.
Math Usage: A vertical translation moves the graph of a relation parallel to
the y-axis.
Use Your Vocabulary
Underline the correct word to complete each sentence.
4. A helicopter takes off vertically / horizontally .
5. A package on a flat conveyor belt moves vertically / horizontally .
6. Stepping side-to-side is a vertical / horizontal movement.
Chapter 6
HSM11A2LC_C06_392-395.indd 392
horizontal
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
vertical (adjective)
392
4/9/09 10:11:28 AM
Key Concepts Families of Radical Functions
Square Root
Radical
Parent function
y 5 !x
n
y 5 !x
Reflection in x-axis
y 5 2!x
n
y 5 2!x
Stretch (a . 1), shrink (0 , a , 1) by factor a
y 5 a!x
n
y 5 a !x
Translation: horizontal by h, vertical by k
y 5 !x 2 h 1 k
n
y 5 !x
2h1k
Problem 1 Translating a Square Root Function Vertically
Got It? What are the graphs of y 5 !x 1 2 and y 5 !x 2 3?
7. The graph of y 5 !x 1 2 is a horizontal / vertical translation of y 5 !x
up / down / left / right 2 units.
8. How does the graph of y 5 !x 2 3 compare to the graph
of y 5 !x ?
y
4
Answers may vary. Sample: It is a vertical
________________________________________________
2
translation of y 5 !x . It is shifted down 3 units.
________________________________________________
________________________________________________
4
2
O
x
2
4
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
2
9. Draw the graph of the function y 5 !x .
Then, use that graph to draw the graphs of
y 5 !x 1 2 and y 5 !x 2 3 on the same grid.
4
Problem 2 Translating a Square Root Function Horizontally
Got It? What are the graphs of y 5 !x 2 3 and y 5 !x 1 2?
10. The graph of y 5 !x 2 3 is a horizontal / vertical translation of y 5 !x
up / down / left / right 3 units.
11. How does the graph of y 5 !x 1 2 compare to the graph of y 5 !x ?
Answers may vary. Sample: It is a horizontal translation of y 5 !x .
_______________________________________________________________________
It is shifted 2 units left.
_______________________________________________________________________
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Lesson 6-8
4/13/09 2:26:46 PM
y
12. Draw the graph of the function y 5 !x. Then, use that graph
to draw the graphs of y 5 !x 2 3 and y 5 !x 1 2 on
the same grid.
2
4
2
O
x
2
4
2
Problem 3
Graphing a Square Root Function
Got It? What is the graph of y 5 3!x 1 2 2 4?
13. Complete the table.
14. Multiplying the y-coordinates of
x
1
4
16
6x
1
2
4
3 6x
3
6
12
y 5 !x by 3 shrinks / stretches the graph.
15. Explain how the graph of y 5 3!x 1 2 2 4
relates to the graph of y 5 !x.
Answers may vary. Sample: It is a
__________________________________________
translation of y 5 !x that is 2 units
__________________________________________
left and 4 units down, and stretched by a
__________________________________________
Problem 4
Solving a Radical Equation by Graphing
Got It? You can model the population of Corpus
16. The graph shows the intersection of P(x) and
y 5 275,000. Draw a vertical line from the point
of intersection to the x-axis.
17. Use the line you drew in Exercise 16 to estimate the
year in which the population of Corpus Christi
was 275,000.
Accept reasonable estimates. Exact answer: 1999
Problem 5
Graphing a Cube Root Function
y
800,000
Population
Christi, TX between the years 1970 and 2005 by the
3
radical function P(x) 5 75000 !x
2 1950. Using this
model, in what year was the population of Corpus
Christi 275,000?
600,000
400,000
200,000
0
x
1920
1950
Year
1980
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
factor of 3.
__________________________________________
3
Got It? What is the graph of y 5 3 2 12 !x
2 2?
Chapter 6
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394
4/9/09 10:11:55 AM
3
18. Write y 5 3 2 12 !x
2 2 in standard form.
3
y 5 212 !x
2213
Underline the correct numbers or words to complete each sentence.
19. The shift is 2 / 3 units left / right and 2 / 3 units up / down .
20. There is a shrink / stretch by a factor of 12 / 2 and the result is / is not reflected.
y
21. Circle the graph of
y532
1 3
2 !x
6
2 2.
4
2
x
2
4 2
3
y 6x
y
2
4
3
y 6x
x
2
2
4 2
4
2
4
Lesson Check • Do you UNDERSTAND?
Error Analysis Your friend states that the graph of g(x) 5 !2x 2 1 is a reflection
of the graph of f (x) 5 2!x 1 1 across the x-axis. Describe your friend’s error.
22. The graph of the function 2f (x) / f (2x) is a reflection of f (x) over the x-axis.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
23. Write the expression that is equal to 2f (x).
!x 1 1
24. Explain the error your friend made.
Answers may vary. Sample: The student multiplied the radicand
_______________________________________________________________________
by 21 instead of multiplying the entire radical by 21.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
radical function
square root function
Rate how well you can graph square root functions.
Need to
review
0
2
4
6
8
Now I
get it!
10
395
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Lesson 6-8
4/9/09 10:12:02 AM