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Splash Screen. F ive-Minut e Check T hen/Now New Voc abulary Key Concept : Operations with Functions Exa mple 1: Operations with Functions Key Concept: Composition of Functions Example 2: Compose Two Functions Example 3 : Find a Composite Function with a Restricted Domain

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  1. Splash Screen

  2. Five-Minute Check Then/Now New Vocabulary Key Concept: Operations with Functions Example 1: Operations with Functions Key Concept: Composition of Functions Example 2: Compose Two Functions Example 3: Find a Composite Function with a Restricted Domain Example 4: Decompose a Composite Function Example 5: Real-World Example: Compose Real-World Functions Lesson Menu

  3. Use the graph of y = x2to describe the graph of the related function y = 0.5x2. A.The parent graph is translated up 0.5 units. B.The parent graph is compressed horizontally by a factor of 0.5. C.The parent graph is compressed vertically by a factor of 0.5. D.The parent graph is translated down 0.5 units. 5–Minute Check 1

  4. Use the graph of y = x2to describe the graph of the related function y = (x – 4)2– 3. A.The parent graph is translated left 3 units and up 4 units. B.The parent graph is translated right 3 units and down 4 units. C.The parent graph is translated left 4 units and down 3 units. D.The parent graph is translated right 4 units and down 3 units. 5–Minute Check 2

  5. A. B. C. D. 5–Minute Check 3

  6. Identify the parent function f(x) if and describe how the graphs of g(x) and f(x) are related. A.f(x) = x; g(x) is f(x) translated left 4 units. B.f(x) = |x|; g(x) is f(x) translated right 4 units. C.g(x) is f(x) translated right 4 units. D.g(x) is f(x) translated left 4 units. 5–Minute Check 4

  7. You evaluated functions. (Lesson 1-1) • Perform operations with functions. • Find compositions of functions. Then/Now

  8. composition Vocabulary

  9. Key Concept 1

  10. The domain of f and g are both so the domain of (f + g) is Answer: Operations with Functions A. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f + g)(x). (f + g)(x) = f(x) + g(x) Definition of sum oftwo functions = (x2 – 2x) + (3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = x2 + x – 4 Simplify. Example 1

  11. The domain of f and h are both so the domain of (f – h) is Answer: Operations with Functions B. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f – h)(x). (f – h)(x) = f(x) – h(x) Definition of difference of two functions = (x2 – 2x) – (–2x2 + 1) f(x) = x2 – 2x; h(x) = –2x2 + 1 = 3x2 – 2x – 1 Simplify. Example 1

  12. The domain of f and g are both so the domain of (f ● g) is Answer: Operations with Functions C. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f ● g)(x). (f ● g)(x) = f(x) ● g(x) Definition of product of two functions = (x2 – 2x)(3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = 3x3 – 10x2 + 8x Simplify. Example 1

  13. D. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for Operations with Functions Definition of quotient of two functions f(x) = x2 – 2x; h(x) = –2x2 + 1 Example 1

  14. The domains of h and f are both (–∞, ∞), but x = 0 or x = 2 yields a zero in the denominator of . So, the domain of (–∞, 0) È (0, 2) È (2, ∞). Answer: D = (–∞, 0) È (0, 2) È (2, ∞) Operations with Functions Example 1

  15. Find (f + g)(x), (f – g)(x), (f ● g)(x), and for f(x) = x2 + x, g(x) = x – 3. State the domain of each new function. Example 1

  16. A. B. C. D. Example 1

  17. Key Concept 2

  18. = 2(x2 + 6x + 9) – 1 Expand (x +3)2 = 2x2 + 12x + 17 Simplify. = f(x + 3) Replace g(x) with x + 3 = 2(x + 3)2 – 1 Substitute x + 3 for x in f(x). Compose Two Functions A. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](x). Answer: [f ○ g](x) = 2x2 + 12x + 17 Example 2

  19. = (2x2 – 1) + 3 = 2x2 + 2 Substitute 2x2 – 1 for x in g(x). Simplify Compose Two Functions B. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [g ○ f](x). Answer: [g ○ f](x) = 2x2 + 2 Example 2

  20. Compose Two Functions C. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](2). Evaluate the expression you wrote in part A for x = 2. [f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x. = 49 Simplify. Answer: [f ○ g](2) = 49 Example 2

  21. Find for f(x) = 2x – 3 and g(x) = 4 + x2. A. 2x2 + 11; 4x2 – 12x + 13; 23 B. 2x2 + 11; 4x2 – 12x + 5; 23 C. 2x2 + 5; 4x2 – 12x + 5; 23 D. 2x2 + 5; 4x2 – 12x + 13; 23 Example 2

  22. A. Find . Find a Composite Function with a Restricted Domain Example 3

  23. To find , you must first be able to find g(x) = (x – 1)2, which can be done for all real numbers. Then you must be able to evaluate for each of these g(x)-values, which can only be done when g(x) > 1. Excluding from the domain those values for which 0 < (x – 1)2 <1, namely when 0 < x < 1, the domain of f ○ g is (–∞, 0] È [2, ∞). Now find [f ○ g](x). Find a Composite Function with a Restricted Domain Example 3

  24. Notice that is not defined for 0 < x < 2. Because the implied domain is the same as the domain determined by considering the domains of f and g, we can write the composition as for (–∞, 0] È [2, ∞). Find a Composite Function with a Restricted Domain Replace g(x) with (x – 1)2. Substitute (x – 1)2 for x in f(x). Simplify. Example 3

  25. Answer: for (–∞, 0] È [2, ∞). Find a Composite Function with a Restricted Domain Example 3

  26. B. Find f ○ g. Find a Composite Function with a Restricted Domain Example 3

  27. To find f ○ g, you must first be able to find , which can be done for all real numbers x such that x2 1. Then you must be able to evaluate for each of these g(x)-values, which can only be done when g(x)  0. Excluding from the domain those values for which 0 >x2 – 1, namely when –1 < x< 1, the domain of f ○ g is (–∞, –1) È (1, ∞). Now find [f ○ g](x). Find a Composite Function with a Restricted Domain Example 3

  28. Find a Composite Function with a Restricted Domain Example 3

  29. Answer: Find a Composite Function with a Restricted Domain Example 3

  30. Check Use a graphing calculator to check this result. Enter the function as . The graph appears to have asymptotes at x = –1 and x = 1. Use the TRACE feature to help determine that the domain of the composite function does not include any values in the interval [–1, 1]. Find a Composite Function with a Restricted Domain Example 3

  31. Find a Composite Function with a Restricted Domain Example 3

  32. Find f ○ g. A. D =(–∞, –1)  (–1, 1)  (1, ∞); B. D =[–1, 1]; C. D =(–∞, –1)  (–1, 1)  (1, ∞); D. D =(0, 1); Example 3

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