Perform Function Operations and Composition

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TEKS 6.3 a.3, 2A.1.A Perform Function Operations and Composition Before You performed operations with algebraic epressions. Now You will perform operations with functions. Why?

These operations can be defined for any number of functions. KEY CONCEPT For Your Notebook Operations on Functions Let f and g be any two functions.

Operation Definition Eample: f() 5 5, g() 5 1 2 Addition h() 5 f() 1 g() h() 5 5 1 ( 1 2) 5 6 1 2 Subtraction h() 5 f() 2 g() h() 5 5 2 ( 1 2) 5 4 2 2 Multiplication h() 5 f() p g() h() 5 5( 1 2) 5 5

1 TEKS 6.3 a.3, 2A.1.A Perform Function Operations and Composition Before You performed operations with algebraic epressions. Now You will perform operations with functions. Why? So you can model biological processes, as in Eample 3. Key Vocabulary power function composition In Chapter 5 you learned how to add, subtract, multiply, and divide polynomial functions. These operations can be defined for any number of functions. KEY CONCEPT For Your Notebook Operations on Functions Let f and g be any two functions. A new function h can be defined by performing any of the four basic operations on f and g. Operation Definition Eample: f() 5 5, g() Addition h() 5 f() 1 g() h() ( 1 2) Subtraction h() 5 f() 2 g() h() ( 1 2) Multiplication h() 5 f() p g() h() 5 5( 1 2) Division h() 5 f() } g() h() 5 5 } 1 2 The domain of h consists of the -values that are in the domains of both f and g. Additionally, the domain of the quotient does not include -values for which g() 5 0. POWER FUNCTIONS So far you have studied several types of functions, including linear functions, quadratic functions, and polynomial functions of higher degree. Another common type of function is a power function, which has the form y 5 a b where a is a real number and b is a rational number. E XAMPLE 1 Add and subtract functions Let f() 5 4 1/2 and g() 529 1/2. Find the following. a. f() 1 g() b. f() 2 g() c. the domains of f 1 g and f 2 g a. f() 1 g() 5 4 1/2 1 (29 1/2 ) 5 [4 1 (29)] 1/ /2 REVIEW DOMAIN For help with domains of functions, see p. 72. b. f() 2 g() 5 4 1/2 2 (29 1/2 ) 5 [4 2 (29)] 1/ /2 c. The functions f and g each have the same domain: all nonnegative real numbers. So, the domains of f 1 g and f 2 g also consist of all nonnegative real numbers. 428 Chapter 6 Rational Eponents and Radical Functions

2 E XAMPLE 2 Multiply and divide functions Let f() 5 6 and g() 5 3/4. Find the following. a. f() p g() b. f() } g() c. the domains of f p g and f } g a. f() p g() 5 (6)( 3/4 ) 5 6 (1 1 3/4) 5 6 7/4 b. f() } g() 5 6 } 3/4 5 6(1 2 3/4) 5 6 1/4 c. The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. So, the domain of f p g consists of all nonnegative real numbers. Because g(0) 5 0, the domain of } f is restricted g to all positive real numbers. E XAMPLE 3 TAKS Solve a REASONING: multi-step problem Multi-Step Problem RHINOS For a white rhino, heart rate r (in beats per minute) and life span s (in minutes) are related to body mass m (in kilograms) by these functions: r(m) 5 241m s(m) 5 ( )m 0.2 Find r(m) p s(m). Eplain what this product represents. STEP 1 Find and simplify r(m) p s(m). r(m) p s(m) 5 241m F( )m 0.2 G Write product of r(m) and s(m) ( )m ( ) Product of powers property 5 ( )m Simplify. STEP 2 Interpret r(m) p s(m). 5 ( )m Use scientific notation. Multiplying heart rate by life span gives the total number of heartbeats for a white rhino over its entire lifetime. GUIDED PRACTICE for Eamples 1, 2, and 3 Let f() 522 2/3 and g() 5 7 2/3. Find the following. 1. f() 1 g() 2. f() 2 g() 3. the domains of f 1 g and f 2 g Let f() 5 3 and g() 5 1/5. Find the following. 4. f() p g() 5. f() } g() 6. the domains of f p g and f } g 7. RHINOS Use the result of Eample 3 to find a white rhino s number of heartbeats over its lifetime if its body mass is kilograms. 6.3 Perform Function Operations and Composition 429

COMPOSITION OF FUNCTIONS Another operation that can be performed with two functions is composition.

Composition of Functions The composition of a function g with a function f is: h() 5 g(f()) The domain of h is the set of all -values such that is in the domain of f and f() is in the domain of g.

What is the value of g(f(4))? A 29 B 21 C 1 D 9 To evaluate g(f(4)), you first must find f(4). f(4) 5 3(4) 2 14 522 Then g(f(4)) 5 g(22) 5 (22) 2 1 5 5 4 1 5 5 9. So, the value of g(f(4)) is 9.

3 COMPOSITION OF FUNCTIONS Another operation that can be performed with two functions is composition. KEY CONCEPT For Your Notebook READING As with subtraction and division of functions, you need to be alert to the order of functions when they are composed. In general, f(g()) is not equal to g( f()). Composition of Functions The composition of a function g with a function f is: h() 5 g(f()) The domain of h is the set of all -values such that is in the domain of f and f() is in the domain of g. Domain of f Input of f Range of f Output of f f () Input of g Domain of g g(f ()) Output of g Range of g E XAMPLE 4 TAKS PRACTICE: Multiple Choice Let f() and g() What is the value of g(f(4))? A 29 B 21 C 1 D 9 To evaluate g(f(4)), you first must find f(4). f(4) 5 3(4) Then g(f(4)) 5 g(22) 5 (22) So, the value of g(f(4)) is 9. c The correct answer is D. A B C D E XAMPLE 5 Find compositions of functions Let f() and g() Find the following. a. f(g()) b. g(f()) c. f(f()) d. the domain of each composition a. f(g()) 5 f(5 2 2) 5 4(5 2 2) } AVOID ERRORS You cannot always determine the domain of a composition from its equation. For instance, the domain of f( f()) 5 appears to be all real numbers, but it is actually all real numbers ecept zero. b. g(f()) 5 g(4 21 ) 5 5(4 21 ) } 2 2 c. f(f()) 5 f (4 21 ) 5 4(4 21 ) (4 21 ) d. The domain of f(g()) consists of all real numbers ecept 5 2 } 5 because g1 2 } is not in the domain of f. (Note that f(0) 5 4 } 0, which is undefined.) The domains of g(f()) and f(f()) consist of all real numbers ecept 5 0, again because 0 is not in the domain of f. 430 Chapter 6 Rational Eponents and Radical Functions

Use composition of functions to do the following: Find the sale price of your purchase when the $10 gift certificate is applied before the 15% discount.

4 E XAMPLE 6 Solve TAKS a REASONING: multi-step problem Multi-Step Problem PAINT STORE You have a $10 gift certificate to a paint store. The store is offering 15% off your entire purchase of any paints and painting supplies. You decide to purchase a $30 can of paint and $25 worth of painting supplies. Use composition of functions to do the following: Find the sale price of your purchase when the $10 gift certificate is applied before the 15% discount. Find the sale price of your purchase when the 15% discount is applied before the $10 gift certificate. STEP 1 STEP 2 STEP 3 Find the total amount of your purchase. The total amount for the paint and painting supplies is $30 1 $25 5 $55. Write functions for the discounts. Let be the regular price, f() be the price after the $10 gift certificate is applied, and g() be the price after the 15% discount is applied. Function for $10 gift certificate: f() Function for 15% discount: g() Compose the functions. The composition g(f()) represents the sale price when the $10 gift certificate is applied before the 15% discount. g(f()) 5 g( 2 10) ( 2 10) The composition f(g()) represents the sale price when the 15% discount is applied before the $10 gift certificate. f(g()) 5 f(0.85) STEP 4 Evaluate the functions g(f()) and f(g()) when g(f(55)) ( ) (45) 5 $38.25 f(g(55)) (55) $36.75 c The sale price is $38.25 when the $10 gift certificate is applied before the 15% discount. The sale price is $36.75 when the 15% discount is applied before the $10 gift certificate. at classzone.com GUIDED PRACTICE for Eamples 4, 5, and 6 Let f() and g() Find the following. 8. g(f(5)) 9. f(g(5)) 10. f(f(5)) 11. g(g(5)) 12. Let f() and g() Find f(g()), g(f()), and f(f()). Then state the domain of each composition. 13. WHAT IF? In Eample 6, how do your answers change if the gift certificate to the paint store is $15 and the store discount is 20%? 6.3 Perform Function Operations and Composition 431

6.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 3, 13, and 45 5 TAKS PRACTICE AND REASONING Es. 11, 38, 39, 44, 48, and 49 5 MULTIPLE REPRESENTATIONS E. 46 1.

WRITING Tell whether the sum of two power functions is sometimes, always, or never a power function. Eplain your reasoning. EXAMPLE 1 on p. 428 for Es.

f() 2 g() 8. g() 2 f() 9. f() 2 f() 10. g() 2 g() 11. MULTIPLE TAKS REASONING CHOICE What is f() 1 g() if f() 527 2/3 2 1 and g() 5 2 2/3 1 6?

5 6.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 3, 13, and 45 5 TAKS PRACTICE AND REASONING Es. 11, 38, 39, 44, 48, and 49 5 MULTIPLE REPRESENTATIONS E VOCABULARY Copy and complete: The function h() 5 g(f()) is called the? of the function g with the function f. 2. WRITING Tell whether the sum of two power functions is sometimes, always, or never a power function. Eplain your reasoning. EXAMPLE 1 on p. 428 for Es ADD AND SUBTRACT FUNCTIONS Let f() 523 1/ /2 and g() 5 5 1/ /2. Perform the indicated operation and state the domain. 3. f() 1 g() 4. g() 1 f() 5. f() 1 f() 6. g() 1 g() 7. f() 2 g() 8. g() 2 f() 9. f() 2 f() 10. g() 2 g() 11. MULTIPLE TAKS REASONING CHOICE What is f() 1 g() if f() 527 2/3 2 1 and g() 5 2 2/3 1 6? A 5 2/3 2 5 B 25 2/3 1 5 C 9 2/3 1 7 D 29 2/3 2 7 EXAMPLE 2 on p. 429 for Es MULTIPLY AND DIVIDE FUNCTIONS Let f() 5 4 2/3 and g() 5 5 1/2. Perform the indicated operation and state the domain. 12. f() p g() 13. g() p f() 14. f() p f() 15. g() p g() 16. f() } g() 17. g() } f() 18. f() } f() 19. g() } g() EXAMPLE 4 on p. 430 for Es EXAMPLE 5 on p. 430 for Es EVALUATE COMPOSITIONS OF FUNCTIONS Let f() , g() 52 2, and h() 5} 2 2. Find the indicated value f(g(23)) 21. g(f(2)) 22. h(f(29)) 23. g(h(8)) 24. h(g(5)) 25. f(f(7)) 26. h(h(24)) 27. g(g(25)) FIND COMPOSITIONS OF FUNCTIONS Let f() , g() , and h() 5} 1 4. Perform the indicated operation and state the domain f(g()) 29. g(f()) 30. h(f()) 31. g(h()) 32. h(g()) 33. f(f()) 34. h(h()) 35. g(g()) ERROR ANALYSIS Let f() and g() 5 4. Describe and correct the error in the composition. 36. f(g()) 5 f(4) 37. g(f()) 5 g( 2 2 3) 5 ( 2 2 3)(4) Chapter 6 Rational Eponents and Radical Functions

38. MULTIPLE TAKS REASONING CHOICE What is g(f()) if f() 5 7 2 and g() 5 3 22? A 3 } 49 4 B 21 C 21 4 D 7 } 9 4 39.

h() 5 4 } 3 2 1 7 42. h() 5 2 1 9 PROBLEM SOLVING EXAMPLE 3 on p. 429 for Es. 43, 46 43.

6 38. MULTIPLE TAKS REASONING CHOICE What is g(f()) if f() and g() ? A 3 } 49 4 B 21 C 21 4 D 7 } OPEN-ENDED TAKS REASONING MATH Find two different functions f and g such that f(g()) 5 g(f()). CHALLENGE Find functions f and g such that f(g()) 5 h(), g() Þ, and f() Þ. 40. h() 5 3 Ï } h() 5 4 } h() PROBLEM SOLVING EXAMPLE 3 on p. 429 for Es. 43, BIOLOGY For a mammal that weighs w grams, the volume b (in milliliters) of air breathed in and the volume d (in milliliters) of dead space (the portion of the lungs not filled with air) can be modeled by: b(w) w d(w) w The breathing rate r (in breaths per minute) of a mammal that weighs w grams can be modeled by: 1.1w0.734 r(w) 5} b(w) 2 d(w) Simplify r(w) and calculate the breathing rate for body weights of 6.5 grams, 300 grams, and 70,000 grams. EXAMPLE 6 on p. 431 for Es SHORT TAKS REASONING RESPONSE The cost (in dollars) of producing sneakers in a factory is given by C() The number of sneakers produced in t hours is given by (t) 5 50t. Find C((t)). Evaluate C((5)) and eplain what this number represents. 45. MULTI-STEP PROBLEM An online movie store is having a sale. You decide to open a charge account and buy four DVDs. DVDs a. Use composition of functions to find the sale price of $85 worth of DVDs when the $15 discount is applied before the 10% discount. b. Use composition of functions to find the sale price of $85 worth of DVDs when the 10% discount is applied before the $15 discount. c. Which order of discounts gives you a better deal? Eplain. 6.3 Perform Function Operations and Composition 433

Write a function r() for the time he spends running from point A to point D. Elvis s swimming speed is about 0.9 meter per second.

7 46. MULTIPLE REPRESENTATIONS A mathematician at a lake throws a tennis ball from point A along the water s edge to point B in the water, as shown. His dog, Elvis, first runs along the beach from point A to point D and then swims to fetch the ball at point B. a. Using a Diagram Elvis s running speed is about 6.4 meters per second. Write a function r() for the time he spends running from point A to point D. Elvis s swimming speed is about 0.9 meter per second. Write a function s() for the time he spends swimming from point D to point B. b. Writing a Function Write a function t() that represents the total time Elvis spends traveling from point A to point D to point B. c. Using a Graph Use a graphing calculator to graph t(). Find the value of that minimizes t(). Eplain the meaning of this value. 47. CHALLENGE To approimate the square root of a number n, the Babylonians used a method that involves starting with an initial guess and calculating a sequence of values that approaches the eact answer. Their method was based on the function shown at the right. a. Let n 5 2, and choose 5 1 as an initial guess for Ï } n 5 Ï } 2. Calculate f(), f(f()), f(f(f())), and f(f(f(f()))). f() 5 1 n } } 2 b. How many times do you need to compose the function in order for the result to approimate Ï } 2 to three decimal places? si decimal places? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 5.1; TAKS Workbook 48. TAKS PRACTICE Which epression is equivalent to (6 3 y 5 z 21 )(23 24 y 2 )? TAKS Obj. 5 A 2 18y10 } 12 z B 2 18z } 7 y 3 C 2 18y7 } z D 2 y7 } 18z REVIEW Lesson 1.5; TAKS Workbook 49. TAKS PRACTICE In a high school marching band, 68% of the members are underclassmen. The rest of the members of the marching band are seniors. Which equation best represents the number of seniors, s, in the band in terms of the total number of students, t, in the band? TAKS Obj. 10 F s 5 8 } 17 t G s 5 8 } 25 t H s 5 17 } 8 t J s 5 25 } 8 t 434 Chapter EXTRA 6 Rational PRACTICE Eponents for and Lesson Radical 6.3, p. Functions 1015 ONLINE QUIZ at classzone.com

Graphing Calculator ACTIVITY Use after Lesson 6.3 6.

com Keystrokes QUESTION How can you use a graphing calculator to perform operations

Find f(4) 1 g(4) and f(g(22)). STEP 1 Form sum Enter y 1 5 2 2 3 1 6 and y 2 5 2 4.

To do so, press, choose the Y-Vars menu, and select Function.

The screen shows that y 3 (4) 5 10, so f(4) 1 g(4) 5 10.

composition f(g()) can be entered as y 3 5 y 1 (y 2 ).

The screen shows that y 3 (22) 5 60, so f(g(22)) 5 60.

8 Graphing Calculator ACTIVITY Use after Lesson Use Operations with Functions TEKS a.3, a.5, a.6 TEXAS classzone.com Keystrokes QUESTION How can you use a graphing calculator to perform operations with functions? EXAMPLE Perform function operations Let f() and g() Find f(4) 1 g(4) and f(g(22)). STEP 1 Form sum Enter y and y The sum can be entered as y 3 5 y 1 1 y 2. To do so, press, choose the Y-Vars menu, and select Function. STEP 2 Evaluate sum On the home screen, enter y 3 (4) and press. The screen shows that y 3 (4) 5 10, so f(4) 1 g(4) Y1=X2-3X+6 Y2=X-4 Y3=Y1+Y2 Y4= Y5= Y6= Y7= Y3(4) 10 STEP 3 Form composition The composition f(g()) can be entered as y 3 5 y 1 (y 2 ). STEP 4 Evaluate composition On the home screen, enter y 3 (22) and press. The screen shows that y 3 (22) 5 60, so f(g(22)) Y1=X2-3X+6 Y2=X-4 Y3=Y1(Y2) Y4= Y5= Y6= Y7= Y3(-2) 60 P RACTICE Use a graphing calculator and the functions f and g to find the indicated value. 1. f() , g() : g(7) 1 f(7) 2. f() 5 1/3, g() 5 9: f(28) } g(28) 3. f() , g() : g(2) 2 f(2) 4. f() , g() 5 2 6: f(g(5)) 6.3 Perform Function Operations and Composition 435

MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons 6.1 6.3 MULTIPLE CHOICE 1. BOWLING The formula for the volume V of a sphere in terms of its surface area S is V 5 3 21 (4π) 21/2 (S 3 ) 1/2.

3 classzone.com 4. SWIMMING POOL A cylindrical above-ground pool has a height of 5 feet and a radius of feet. You use a hose to fill the pool with water.

AREA OF SHADED REGION A triangle is inscribed in a square, as shown. Which function r() represents the area of the shaded region? TEKS 2A

3 A f() 5 3 22 B f() 5 1 3 C f() 5 5 2 D f() 5 1/2 J r() 5 1 } 2 4 3. SALARY You are working as a sales representative for a clothing manufacturer.

03 Which epression represents your bonus when > 100,000? TEKS a.3 A B C D f() p g() f() } g() f(g()) g(f()) 6.

9 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons MULTIPLE CHOICE 1. BOWLING The formula for the volume V of a sphere in terms of its surface area S is V (4π) 21/2 (S 3 ) 1/2. A candlepin bowling ball has a surface area of about 79 square inches. What is its volume to the nearest cubic inch? TEKS 2A.2.A A 66 in. 3 B 184 in. 3 C 368 in. 3 D 594 in. 3 classzone.com 4. SWIMMING POOL A cylindrical above-ground pool has a height of 5 feet and a radius of feet. You use a hose to fill the pool with water. Water flows from the hose at a rate of 128 cubic feet per hour. After 8.8 hours, the pool is half full. What is the radius of the pool to the nearest foot? Use 3.14 for π. TEKS 2A.2.A 2. AREA OF SHADED REGION A triangle is inscribed in a square, as shown. Which function r() represents the area of the shaded region? TEKS 2A.2.A 1 2 F 6feet G 7feet H 12 feet J 24 feet F r() 5 } 3 4 G r() 5 } H r() 5 } FUNCTION COMPOSITION Which function f() satisfies the condition that f(f()) 5? TEKS a.3 A f() B f() C f() D f() 5 1/2 J r() 5 1 } SALARY You are working as a sales representative for a clothing manufacturer. You are paid an annual salary plus a bonus of 3% of your sales over $100,000. Consider these two functions: f() ,000 g() Which epression represents your bonus when > 100,000? TEKS a.3 A B C D f() p g() f() } g() f(g()) g(f()) 6. SIMPLIFYING AN EXPRESSION What is the simplified form of the epression 1} 161/2 4 1/2 2 5? TEKS 2A.2.A F 2 G 32 H 512 J 1024 GRIDDED ANSWER GEOMETRY The volume of a sphere is 900 cubic inches. Use the formula for the volume of a sphere, V 5 } 4 πr 3, to find the 3 radius r of the sphere to the nearest hundredth of an inch. Use 3.14 for π. TEKS 2A.2.A 436 Chapter 6 Rational Eponents and Radical Functions

Investigating g Algebra ACTIVITY Use before Lesson 6.4 6.4 Eploring Inverse Functions MATERIALS graph paper straightedge TEKS a.3, a.5, a.6, 2A.4.C QUESTION How are a function and its inverse related?

STEP 2 Interchange coordinates Interchange the - and y-coordinates of the ordered pairs found in Step 1. Plot the new points and draw the line that passes through them.

How are the graphs geometrically related? STEP 5 Describe functions In words, f is the function that subtracts 3 from and then divides the result by 2. Describe the function g in words.

DRAW CONCLUSIONS Use your observations to complete these eercises Complete Eercises 1 3 for each function below. f() 5 3 1 2 f() 5 2 1 } 6 f() 5 4 2 3 } 2 1.

10 Investigating g Algebra ACTIVITY Use before Lesson Eploring Inverse Functions MATERIALS graph paper straightedge TEKS a.3, a.5, a.6, 2A.4.C QUESTION How are a function and its inverse related? EXPLORE Find the inverse of f() } 2 STEP 1 Graph function Choose values of and find the corresponding values of y 5 f(). Plot the points and draw the line that passes through them. STEP 2 Interchange coordinates Interchange the - and y-coordinates of the ordered pairs found in Step 1. Plot the new points and draw the line that passes through them. STEP 3 Write equation Write an equation of the line from Step 2. Call this function g. STEP 4 Compare graphs Fold your graph paper so that the graphs of f and g coincide. How are the graphs geometrically related? STEP 5 Describe functions In words, f is the function that subtracts 3 from and then divides the result by 2. Describe the function g in words. STEP 6 Find compositions Predict what the compositions f(g()) and g(f()) will be. Confirm your predictions by finding f(g()) and g(f()). The functions f and g are called inverses of each other. DRAW CONCLUSIONS Use your observations to complete these eercises Complete Eercises 1 3 for each function below. f() f() } 6 f() } 2 1. Complete Steps 1 3 above to find the inverse of the function. 2. Complete Step 4. How can you graph the inverse of a function without first finding ordered pairs (, y)? 3. Complete Steps 5 and 6. How can you test to see if the function you found in Eercise 1 is indeed the inverse of the original function? 6.4 Use Inverse Functions 437

Find Rational Zeros. has integer coefficients, then every rational zero of f has the following form: x 1 a 0. } 5 factor of constant term a 0

Find Rational Zeros. has integer coefficients, then every rational zero of f has the following form: x 1 a 0. } 5 factor of constant term a 0 .6 Find Rational Zeros TEKS A.8.B; P..D, P..A, P..B Before You found the zeros of a polnomial function given one zero. Now You will find all real zeros of a polnomial function. Wh? So ou can model manufacturing