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## Performing Function Operations 55 Exercises

**Vocabulary and Core Concept Check**

Question 1.**WRITING **Let f and g be any two functions. Describe how you can use f, g, and the four basic operations to create new functions.Answer:

**WRITING **What x-values are not included in the domain of the quotient of two functions?Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 36, find and and state the domain of each. Then evaluate f + g and f g for the given value of x.**

Question 3.f = \, g = 19\ x = 16Answer:

f = \, g = 11\ x = 4Answer:

f = 6x 4×2 7×3, g = 9×2 5x x = 1Answer:

f = 11x + 2×2, g = 7x 3×2 + 4 x = 2Answer:

**In Exercises 712, find and ) and state the domain of each. Then evaluate fg and \ for the given value of x.**

Question 7.f = 2×3, g = \ x = 27Answer:

f = x4, g = \ x = 4Answer:

f = 4x, g = 9×1/2 x = 9Answer:

f = 11×3, g = 7×7/3 x = 8Answer:

f = 7×3/2, g =14×1/3 x = 64Answer:

f = 4×5/4, g = 2×1/2 x = 16Answer:

**USING TOOLS In Exercises 1316, use a graphing calculator to evaluate , , , and ) when x = 5. Round your answers to two decimal places.**

Question 13.

Answer:

Question 19.**MODELING WITH MATHEMATICS** From 1990 to 2010, the numbers of female F and male M employees from the ages of 16 to 19 in the United States can be modeled by F =0.007t2 + 0.10t + 3.7 and M = 0.0001t3 0.009t2 + 0.11t + 3.7, where t is the number of years since 1990.a. Find .b. Explain what represents.Answer:

**Solve the literal equation for n. **

Question 28.

\ = 7bAnswer:

Question 32., , , Answer:

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## Lesson 55 Performing Function Operations

**Essential Question**How can you use the graphs of two functions to sketch the graph of an arithmetic combination of the two functions?Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to form other functions. For example, the functions f = 2x 3 and g = x2 1 can be combined to form the sum, difference, product, or quotient of f and g.f + g = + = x2 + 2x 4 sumf g = = x2 + 2x 2 differencef g = = 2×3 3×2 2x + 3 product

**EXPLORATION 1Graphing the Sum of Two FunctionsWork with a partner.** Use the graphs of f and g to sketch the graph of f + g. Explain your steps.Sample Choose a point on the graph of g. Use a compass or a ruler to measure its distance above or below the x-axis. If above, add the distance to the y-coordinate of the point with the same x-coordinate on the graph of f. If below, subtract the distance. Plot the new point. Repeat this process for several points. Finally, draw a smooth curve through the new points to obtain the graph of f + g.

**Communicate Your Answer**

Question 2.How can you use the graphs of two functions to sketch the graph of an arithmetic combination of the two functions?

Question 3.Check your answers in Exploration 1 by writing equations for f and g, adding the functions, and graphing the sum.

**5.5 Lesson**

**Monitoring Progress**

Question 4.In Example 5, explain why you can evaluate , , and but not ).

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## Nth Roots And Rational Exponents 51 Exercises

**Vocabulary and Core Concept Check**

Question 1.**VOCABULARY** Rewrite the expression a^\) in radical form. Then state the index of the radical.Answer:

Question 2.**COMPLETE THE SENTENCE** For an integer n greater than 1, if bn = a, then bis a ___________ of a.Answer:

Question 3.**WRITING** Explain how to use the sign of a to determine the number of real fourth roots of a and the number of real fifth roots of a.Answer:

Question 4.**WHICH ONE DOESNT BELONG? **Which expression does not belong with the other three? Explain your reasoning.Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 510, find the indicated real nth root of a.**

Question 5.

\x4 = 27Answer:

\x3 = 36Answer:

Question 45.**MODELING WITH MATHEMATICS **When the average price of an item increases from p1 to p2 over a period of n years, the annual rate of inflation r is given by r = \^\) 1. Find the rate of inflation for each item in the table.Answer:

Question 46.**HOW DO YOU SEE IT?** The graph of y = xn is shown in red. What can you conclude about the value of n? Determine the number of real nth roots of a. Explain your reasoning.Answer:

**NUMBER SENSE** Between which two consecutive integers does \ lie? Explain your reasoning.Answer:

**Simplify the expression. Write your answer using only positive exponents. **

Question 51.

Lesson 5.2 Properties of Rational Exponents and Radicals

**Essential Question**How can you use properties of exponents to simplify products and quotients of radicals?

**Communicate Your Answer**

**5.2 Lesson**

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## Lesson 51 Nth Roots And Rational Exponents

**Essential Question**

How can you use a rational exponent to represent a power involving a radical?Previously, you learned that the nth root of a can be represented as\ = \ Definition of rational exponentfor any real number a and integer n greater than 1.

**EXPLORATION 1Exploring the Definition of a Rational ExponentWork with a partner. **Use a calculator to show that each statement is true.a. \ = \b. \ = \c. \ = \d. \ = \e. \ = \f. \ = \

**EXPLORATION 2Writing Expressions in Rational Exponent Form****Work with a partner. **Use the definition of a rational exponent and the properties of exponents to write each expression as a base with a single rational exponent. Then use a calculator to evaluate each expression. Round your answer to two decimal places.a. \^\)b. \^\)c. \^\)d. \^\)e. \^\)f. \^\)

**EXPLORATION 3Writing Expressions in Radical FormWork with a partner.** Use the properties of exponents and the definition of a rational exponent to write each expression as a radical raised to an exponent. Then use a calculator to evaluate each expression. Round your answer to two decimal places.Sample \ = \)2 = \))2 2.92a. \

3 = 14

Question 17.**WHAT IF? **In Example 5, what is the annual depreciation rate when the salvage value is $6000?

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6x + 7xy = 15

Question 13.**ABSTRACT REASONING** Is the order in which you apply properties of exponents important? Explain your reasoning.

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