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Warm Up

Perform Function Operations and Composition. Warm Up. Lesson Presentation. Lesson Quiz. 1. 9 x – 2 x. 7 x. 2. 4. 2. 2. 5. 3. 3. 3. 28 x 4 x 1/5. 2. 7 x. Warm-Up. Perform the indicated operation. Assume all variables are positive. ANSWER. ANSWER.

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Warm Up

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  1. Perform Function Operations and Composition Warm Up Lesson Presentation Lesson Quiz

  2. 1. 9x –2x 7x 2 4 2 2 5 3 3 3 28x 4x1/5 2. 7x Warm-Up Perform the indicated operation. Assume all variables are positive. ANSWER ANSWER Let f(x) = 3x + 5 and g(x) = 2x2 –7. Find the following. g (–4) 4. 3.f (–6) 25 ANSWER ANSWER –13

  3. a. f(x) + g(x) f(x) – g(x) b. Example 1 Letf (x)= 4x1/2andg(x)=–9x1/2. Find the following. SOLUTION f (x) + g(x) = 4x1/2 + (–9x1/2) = [4 + (–9)]x1/2 = –5x1/2 SOLUTION f (x) – g(x) = [4 – (–9)]x1/2 = 13x1/2 = 4x1/2 – (–9x1/2)

  4. The functions fand geach have the same domain: all nonnegative real numbers. So, the domains of f + gand f – galso consist of all nonnegative real numbers. c. the domains of f + gand f – g Example 1 SOLUTION

  5. a. f (x) g(x) b. f (x) f (x) 6x x3/4 g(x) g(x) f (x) g(x) = = 6x1/4 = 6x(1 – 3/4) Example 2 Let f (x)= 6xand g(x) = x3/4. Find the following. SOLUTION = (6x)(x3/4) = 6x(1 + 3/4) = 6x7/4 SOLUTION

  6. The domain of f consists of all real numbers, and the domain of gconsists of all nonnegative real numbers. So, the domain of f gconsists of all nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers. f g Example 2 f g and the domains of f c. g SOLUTION

  7. r(m) s(m) (6 106)m0.2 = 241m–0.25 = • Findr(m) s(m). Example 3 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: • Explain what this product represents.

  8. Find and simplify r(m) s(m). r(m) s(m) = 241m –0.25 [ (6 106)m0.2 ] 241(6 106)m(–0.25 + 0.2) = (1446 106)m –0.05 = (1.446 109)m –0.05 = Example 3 SOLUTION STEP 1 Write product of r(m) and s(m). Product of powers property Simplify. Use scientific notation.

  9. Interpret r(m) s(m). Example 3 STEP 2 Multiplying heart rate by life span gives the total number of heartbeats for a white rhino over its entire lifetime.

  10. the domains of f + gand f – g f (x) + g(x) f (x) – g(x) Guided Practice Let f (x) = –2x2/3andg(x) = 7x2/3. Find the following. f (x) + g(x) = –2x2/3 + 7x2/3 = 5x2/3 SOLUTION = (–2 + 7)x2/3 f (x) – g(x) = –2x2/3 – 7x2/3 SOLUTION = [–2 + ( –7)]x2/3 = –9x2/3 SOLUTION all real numbers; all real numbers

  11. the domains off g and f (x) g(x) f (x) f g g(x) Guided Practice Let f (x) = 3xandg(x) = x1/5. Find the following. SOLUTION 3x6/5 SOLUTION 3x4/5 SOLUTION all real numbers; all real numbers except x = 0.

  12. Use the result of Example 3 to find a white rhino’s number of heartbeats over its lifetime if its body mass is 1.7 105kilograms. about 7.92 108 heartbeats Guided Practice Rhinos SOLUTION

  13. Example 4 SOLUTION To evaluate g(f(3)), you first must find f(3). f(3) = 2(3) – 7 =–1 Theng( f(3))=g(–1) = (–1)2+ 4 = 1 + 4 = 5. So, the value of g(f(3)) is 5. ANSWER The correct answer is D.

  14. 20 g(f(x)) a. f(g(x)) b. – 2 4 x 5x – 2 = f(g(x)) = f(5x– 2) g(f(x)) = Example 5 Let f(x) = 4x–1 and g(x) = 5x – 2. Find the following. SOLUTION = 4(5x– 2)–1 SOLUTION = g(4x–1) = 20x–1 – 2 = 5(4x–1) – 2

  15. c. f(f(x)) Example 5 SOLUTION f(f(x)) = f(4x–1) = 4(4x–1)–1 = 4(4–1x) = 40x = x

  16. ( ) ( ) 2 2 4 5 5 0 Example 5 d. The domain of f(g(x)) consists of all real numbers except x= because g = 0 is not in the domain of f. (Note that f(0) = , which is undefined.) The domains of g(f(x)) and f(f(x)) consist of all real numbers except x = 0, again because 0 is not in the domain of f.

  17. Find the sale price of your purchase when the 15% discount is applied before the $10 gift certificate. • • Find the sale price of your purchase when the $10 gift certificate is applied before the 15% discount. Example 6 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:

  18. Example 6 SOLUTION STEP 1 Find: the total amount of your purchase. The total amount for the paint and painting supplies is $30 + $25 = $55. STEP 2 Write: functions for the discounts. Let x be the regular price, f(x) be the price after the $10 gift certificate is applied, and g(x) be the price after the 15% discount is applied. Function for $10 gift certificate: f (x) = x – 10 Function for 15% discount: g(x) = x – 0.15x = 0.85x

  19. Example 6 STEP 3 Compose the functions. The composition g(f(x)) represents the sale price when the $10 gift certificate is applied before the 15% discount. g(f (x)) = g(x – 10) = 0.85(x – 10) The composition f(g(x)) represents the sale price when the 15% discount is applied before the $10 gift certificate. f (g(x)) = f(0.85x) = 0.85x – 10

  20. ANSWER 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. Example 6 STEP 4 Evaluate: the functions g(f(x)) andf(g(x)) when x = 55. g(f(55)) = 0.85(55– 10) = 0.85(45) = $38.25 = 0.85(55) – 10 = 46.75 – 10 f(g(55))

  21. f(g(5)) f(f(5)) g(g(5)) g(f(5)) Guided Practice Let f(x) = 3x – 8 and g(x) = 2x2. Find the following. SOLUTION 98 SOLUTION 142 SOLUTION 13 SOLUTION 5000

  22. Let f(x) = 2x–1 and g(x) = 2x + 7. Find f(g(x)),g(f(x)), and f(f(x)). Then state the domain of each composition. SOLUTION 2 all real numbers except x = –3.5; 2x + 7 4 + 7 all real numbers except x = 0; x x all real numbers except x = 0. Guided Practice

  23. What If?In Example 6, how do your answers change if the gift certificate to the paint storeis $15 and the store discount is 20%? Guided Practice SOLUTION The sale price is $32 when the $15 gift certificate is applied before the 20% discount. The sale price is $29 when the 20% discount is applied before the $15 gift certificate.

  24. 4x1/4 ANSWER 8x1/4 ANSWER 3.f(x) g(x) ANSWER –12x1/2 all nonnegative real numbers (or x0)  ANSWER 4.the domain offg g(x) 5. –3 ANSWER f(x) All positive real numbers f ANSWER 6.the domain of g Lesson Quiz Letf(x) = –2x1/4 andg(x) = 6x1/4. Find the following. 1.f(x) + g(x) 2.g(x) – f(x)

  25. –167 ANSWER –5x2 +7 ANSWER after ANSWER Lesson Quiz Letf(x) = 5x–3andg(x) = –x2 + 2. Find the following. 7.g( f (–2)) 8.f(g(x)) 9. You have a $25 gift certificate for a book store. The store is offering 10% off all books. When should you use the certificate, before or after the 10% discount?

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