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Splash Screen. Five-Minute Check (over Chapter 5) CCSS Then/Now New Vocabulary Key Concept: Operations on Functions Example 1: Add and Subtract Functions Example 2: Multiply and Divide Functions Key Concept: Composition of Functions Example 3: Compose Functions

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 5) CCSS Then/Now New Vocabulary Key Concept: Operations on Functions Example 1: Add and Subtract Functions Example 2: Multiply and Divide Functions Key Concept: Composition of Functions Example 3: Compose Functions Example 4: Real-World Example: Use Composition of Functions Lesson Menu

  3. Estimate the x-coordinates at which the relative maxima and relative minima occur for the graph of f(x) = x4 + 16x2 – 25. A. 0 B. 1 C. 4 D. 5 5-Minute Check 1

  4. Estimate the x-coordinates at which the relative maxima and relative minima occur for the graph of f(x) = x4 + 16x2 – 25. A. 0 B. 1 C. 4 D. 5 5-Minute Check 1

  5. Solve p3 – 2p2 – 8p = 0. A. –2, 2 B. –2, 0, 4 C. 0, 2, 6 D. 2, 4, 6 5-Minute Check 2

  6. Solve p3 – 2p2 – 8p = 0. A. –2, 2 B. –2, 0, 4 C. 0, 2, 6 D. 2, 4, 6 5-Minute Check 2

  7. A. B. C. D. Solve x4 – 7x2 + 12 = 0. 5-Minute Check 3

  8. A. B. C. D. Solve x4 – 7x2 + 12 = 0. 5-Minute Check 3

  9. A.±15 B. C. D. Which is not a possible rational zero for the function f(x) = 3x3 – 12x2 – 5x + 15? 5-Minute Check 4

  10. A.±15 B. C. D. Which is not a possible rational zero for the function f(x) = 3x3 – 12x2 – 5x + 15? 5-Minute Check 4

  11. A. B.1 C.4 D.13 Which of the following could not be a zero of the function g(x) = 4x3 – 5x + 26? 5-Minute Check 5

  12. A. B.1 C.4 D.13 Which of the following could not be a zero of the function g(x) = 4x3 – 5x + 26? 5-Minute Check 5

  13. Content Standards F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). F.BF.1.b Combine standard function types using arithmetic operations. Mathematical Practices 2 Reason abstractly and quantitatively. CCSS

  14. You performed operations on polynomials. • Find the sum, difference, product, and quotient of functions. • Find the composition of functions. Then/Now

  15. composition of functions Vocabulary

  16. Concept

  17. Add and Subtract Functions A. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f + g)(x). (f + g)(x) = f(x) + g(x) Addition of functions = (3x2 + 7x) + (2x2 – x – 1) f(x) = 3x2 + 7x andg(x) = 2x2 – x – 1 = 5x2 + 6x – 1Simplify. Answer: Example 1

  18. Add and Subtract Functions A. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f + g)(x). (f + g)(x) = f(x) + g(x) Addition of functions = (3x2 + 7x) + (2x2 – x – 1) f(x) = 3x2 + 7x andg(x) = 2x2 – x – 1 = 5x2 + 6x – 1Simplify. Answer: 5x2 + 6x – 1 Example 1

  19. Add and Subtract Functions B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f – g)(x). (f – g)(x) = f(x) – g(x) Subtraction of functions = (3x2 + 7x) – (2x2 – x – 1) f(x) = 3x2 + 7x andg(x) = 2x2 – x – 1 = x2 + 8x + 1Simplify. Answer: Example 1

  20. Add and Subtract Functions B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f – g)(x). (f – g)(x) = f(x) – g(x) Subtraction of functions = (3x2 + 7x) – (2x2 – x – 1) f(x) = 3x2 + 7x andg(x) = 2x2 – x – 1 = x2 + 8x + 1Simplify. Answer:x2 + 8x + 1 Example 1

  21. A. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4, find (f + g)(x). A. 5x2 + 8x – 2 B. 5x2 + 8x + 6 C.x2 – 2x – 6 D. 5x4 + 8x2 – 2 Example 1

  22. A. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4, find (f + g)(x). A. 5x2 + 8x – 2 B. 5x2 + 8x + 6 C.x2 – 2x – 6 D. 5x4 + 8x2 – 2 Example 1

  23. B. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4, find (f – g)(x). A. –x2 + 2x + 5 B.x2 – 2x – 6 C. –x2 + 2x – 2 D. –x2 + 2x + 6 Example 1

  24. B. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4, find (f – g)(x). A. –x2 + 2x + 5 B.x2 – 2x – 6 C. –x2 + 2x – 2 D. –x2 + 2x + 6 Example 1

  25. Multiply and Divide Functions A. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find (f ● g)(x). (f● g)(x) = f(x) ● g(x) Product of functions = (3x2 – 2x + 1)(x – 4) Substitute. = 3x2(x – 4) – 2x(x – 4) + 1(x – 4) Distributive Property = 3x3 – 12x2 – 2x2 + 8x + x – 4 Distributive Property = 3x3 – 14x2 + 9x – 4 Simplify. Answer: Example 2

  26. Multiply and Divide Functions A. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find (f ● g)(x). (f● g)(x) = f(x) ● g(x) Product of functions = (3x2 – 2x + 1)(x – 4) Substitute. = 3x2(x – 4) – 2x(x – 4) + 1(x – 4) Distributive Property = 3x3 – 12x2 – 2x2 + 8x + x – 4 Distributive Property = 3x3 – 14x2 + 9x – 4 Simplify. Answer: 3x3 – 14x2 + 9x – 4 Example 2

  27. B. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find Multiply and Divide Functions Division of functions f(x) = 3x2 – 2x + 1 and g(x) = x – 4 Answer: Example 2

  28. B. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find Answer: Multiply and Divide Functions Division of functions f(x) = 3x2 – 2x + 1 and g(x) = x – 4 Example 2

  29. Since 4 makes the denominator 0, it is excluded from the domain of Multiply and Divide Functions Example 2

  30. A. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find (f ● g)(x). A. 2x3 + 3x2 – x + 2 B. 2x3 + 3x – 2 C. 2x3 + 7x2 + 5x – 2 D. 2x3 + 7x2 + 7x + 2 Example 2

  31. A. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find (f ● g)(x). A. 2x3 + 3x2 – x + 2 B. 2x3 + 3x – 2 C. 2x3 + 7x2 + 5x – 2 D. 2x3 + 7x2 + 7x + 2 Example 2

  32. B. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find. A. B. C. D. Example 2

  33. B. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find. A. B. C. D. Example 2

  34. Concept

  35. Compose Functions A. If f(x) = (2, 6), (9, 4), (7, 7), (0, –1) and g(x) = (7, 0), (–1, 7), (4, 9), (8, 2), find [f ○ g](x) and [g ○ f](x). To find f○g, evaluate g(x) first. Then use the range of g as the domain of f and evaluate f(x). f[g(7)] = f(0) or –1 g(7) = 0 f[g(–1)] = f(7) or 7 g(–1) = 7 f[g(4)] = f(9) or 4 g(4) = 9 f[g(8)] = f(2) or 6 g(8) = 2 Answer: Example 3

  36. Compose Functions A. If f(x) = (2, 6), (9, 4), (7, 7), (0, –1) and g(x) = (7, 0), (–1, 7), (4, 9), (8, 2), find [f ○ g](x) and [g ○ f](x). To find f○g, evaluate g(x) first. Then use the range of g as the domain of f and evaluate f(x). f[g(7)] = f(0) or –1 g(7) = 0 f[g(–1)] = f(7) or 7 g(–1) = 7 f[g(4)] = f(9) or 4 g(4) = 9 f[g(8)] = f(2) or 6 g(8) = 2 Answer: f○g = {(7, –1), (–1, 7), (4, 4), (8, 6)} Example 3

  37. Compose Functions To find g○f, evaluate f(x) first. Then use the range of f as the domain of g and evaluate g(x). g[f(2)] = g(6) g(6) is undefined. g[f(9)] = g(4) or 9 f(9) = 4 g[f(7)] = g(7) or 0 f(7) = 7 g[f(0)] = g(–1) or 7 f(0) = –1 Answer: Example 3

  38. Compose Functions To find g○f, evaluate f(x) first. Then use the range of f as the domain of g and evaluate g(x). g[f(2)] = g(6) g(6) is undefined. g[f(9)] = g(4) or 9 f(9) = 4 g[f(7)] = g(7) or 0 f(7) = 7 g[f(0)] = g(–1) or 7 f(0) = –1 Answer: Since 6 is not in the domain of g, g○f is undefined for x = 2.g○f = {(9, 9), (7, 0), (0, 7)} Example 3

  39. Compose Functions B. Find [f ○ g](x) and [g ○ f](x) for f(x) = 3x2 – x + 4 and g(x) = 2x – 1. State the domain and range for each combined function. [f ○ g](x) = f[g(x)] Composition of functions = f(2x – 1) Replace g(x) with 2x – 1. = 3(2x – 1)2 – (2x – 1) + 4 Substitute 2x – 1 for x in f(x). Example 3

  40. Compose Functions = 3(4x2 – 4x + 1) – 2x + 1 + 4 Evaluate (2x – 1)2. = 12x2 – 14x + 8 Simplify. [g ○ f](x) = g[f(x)] Composition of functions = g(3x2 – x + 4) Replace f(x) with 3x2 – x + 4. Example 3

  41. Compose Functions = 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 for x in g(x). = 6x2 – 2x + 7 Simplify. Answer: Example 3

  42. Compose Functions = 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 for x in g(x). = 6x2 – 2x + 7 Simplify. Answer: So, [f ○ g](x) = 12x2 – 14x + 8; D = {all real numbers}, R = {y│y > 3.91}; and [g ○ f](x) = 6x2 – 2x + 7; D = {all real numbers}, R = {y│y > 6.33}. Example 3

  43. A. If f(x) = {(1, 2), (0, –3), (6, 5), (2, 1)} and g(x) = {(2, 0), (–3, 6), (1, 0), (6, 7)}, find f ○ g and g ○ f. A. f ○ g = {(2, –3), (–3, 5), (1, –3)};g ○ f = {(1, 0), (0, 6), (2, 0)} B.f ○ g = {(1, 0), (0, 6), (2, 0)};g ○ f = {(2, –3), (–3, 5), (1, –3)} C.f ○ g = {(–3, 2), (5, –3), (–3, 1)};g ○ f = {(0, 1), (6, 0), (0, 2)} D.f ○ g = {(0, 1), (6, 0), (0, 2)};g ○ f = {(–3, 2), (5, –3), (–3, 1)} Example 3

  44. A. If f(x) = {(1, 2), (0, –3), (6, 5), (2, 1)} and g(x) = {(2, 0), (–3, 6), (1, 0), (6, 7)}, find f ○ g and g ○ f. A. f ○ g = {(2, –3), (–3, 5), (1, –3)};g ○ f = {(1, 0), (0, 6), (2, 0)} B.f ○ g = {(1, 0), (0, 6), (2, 0)};g ○ f = {(2, –3), (–3, 5), (1, –3)} C.f ○ g = {(–3, 2), (5, –3), (–3, 1)};g ○ f = {(0, 1), (6, 0), (0, 2)} D.f ○ g = {(0, 1), (6, 0), (0, 2)};g ○ f = {(–3, 2), (5, –3), (–3, 1)} Example 3

  45. B. Find [f ○ g](x) and [g ○ f](x) for f(x) = x2 + 2x + 3 and g(x) = x + 5. State the domain and range for each combined function. A. [f ○ g](x) = x2 + 2x + 8; D = {all real numbers}, R = {y│y ≥ 2}; and [g ○ f](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 7} B. [f ○ g](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 7}; and [g ○ f](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 2} C. [f ○ g](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 2}; and [g ○ f](x) = x2 + 2x + 8; D = {all real numbers}, R = {y│y ≥ 7} Example 3

  46. B. Find [f ○ g](x) and [g ○ f](x) for f(x) = x2 + 2x + 3 and g(x) = x + 5. State the domain and range for each combined function. A. [f ○ g](x) = x2 + 2x + 8; D = {all real numbers}, R = {y│y ≥ 2}; and [g ○ f](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 7} B. [f ○ g](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 7}; and [g ○ f](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 2} C. [f ○ g](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 2}; and [g ○ f](x) = x2 + 2x + 8; D = {all real numbers}, R = {y│y ≥ 7} Example 3

  47. Use Composition of Functions TAXESHector has $100 deducted from every paycheck for retirement. He can have this deduction taken before state taxes are applied, which reduces his taxable income. His state income tax is 4%. If Hector earns $1500 every pay period, find the difference in his net income if he has the retirement deduction taken before or after state taxes. Understand Let x = his income per paycheck, r(x) = his income after the deduction for retirement, and t(x) = his income after the deduction for state income tax. Example 4

  48. Use Composition of Functions Plan Write equations for r(x) and t(x). $100 is deducted for retirement. r(x) = x – 100 The tax rate is 4%. t(x) = x – 0.04x Solve If Hector has his retirement deducted before taxes, then his net income is represented by [t ○ r](1500). [t ○ r](1500) = t(1500 – 100) Replace x with 1500 inr(x) = x – 100. = t(1400) Example 4

  49. Use Composition of Functions = 1400 – 0.04(1400) Replace x with 1400 in t(x) = x – 0.04x. = 1344 If Hector has his retirement deducted after taxes, then his net income is represented by [r○t](1500). Replace x with 1500 int(x) = x – 0.04x. [r○t](1500) = r[1500 – 0.04(1500)] = r(1500 – 60) = r(1440) Example 4

  50. Use Composition of Functions = 1440 – 100 Replace x with 1440 in r(x) = x – 100. = 1340 Answer: Example 4

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