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1.6 Composition of Functions Tues Sept 24

1.6 Composition of Functions Tues Sept 24. Do Now Find the domain of . HW Review: p.131 #39-49 59-63. 39-49) Graph in book 59) D: (- inf , inf ) R: (-1, inf ) f(x) = -2 for x < 2 -5 for x = 2 4 for x > 2 61) D: (- inf , inf ) R: (- inf , 1] U [2, inf ) g (x) = x for x <= -1

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1.6 Composition of Functions Tues Sept 24

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  1. 1.6Composition of FunctionsTues Sept 24 Do Now Find the domain of

  2. HW Review: p.131 #39-49 59-63 • 39-49) Graph in book • 59) D: (-inf, inf) R: (-1, inf) f(x) = -2 for x < 2 -5 for x = 2 4 for x > 2 • 61) D: (-inf, inf) R: (-inf, 1] U [2, inf) g(x) = x for x <= -1 2 for -1 < x < 2 x for x >= 2

  3. 63 • 63) D: [-5, 3) R: (-3, 5) h(x) = x + 8 for -5 <= x < -3 3 for -3 <= x <= 1 3x – 6 for 1 < x <= 3

  4. Composite Functions • A composite function is a function whose range is determined by the output of another function • One function is composed in another function • Ex: The DO NOW function is a square root function composed in a rational function

  5. Composition of Functions • The composite function , the composition of f and g, is defined as where x is in the domain of g, and g(x) is in the domain of f Note: This means only the range of g(x) can be plugged into f(x)!

  6. Ex • Given that f(x) = 2x – 5 and g(x) = x^2 – 3x + 8, find f(g(x)) and g(f(x))

  7. Compositions and Domains • When considering the domain of a composite function, you must look at the following: • The domain of the function that is being ‘plugged in’ • The domain of the final composition Note: Just because a composition simplifies into something with no restrictions does not mean the domain has no restrictions!

  8. Ex • Given that and g(x) = x – 3, find f(g(x)) and g(f(x)) and determine the domains of each

  9. Ex • Given that f(x) = 1 / (x – 2) and g(x) = 5/x, find f(g(x)) and g(f(x)), and the domain of each

  10. Closure • Given f(x) = 3x – 2 and g(x) = x^2 + 5, find f(g(x)) and g(f(x)) • HW: p.144 #69-87 odds • Check your answers! I don’t put up the answers anymore!

  11. 1.6 Decomposing CompositionsWed Sept 25 • Do Now • Given f(x) = sqrt x and g(x) = 2 – 3x, find f(g(x)) and g(f(x)) and their domains

  12. HW: p.144 #69-87 odds • Check your book

  13. Decomposing a Function • In calculus we’ll need to identify which function is the ‘inside’ and the ‘outside’ • This method is called decomposing

  14. Ex • If h(x) = (2x – 3)^5, find f(x) and g(x) such that h(x) = f(g(x))

  15. Ex • If , find f(x) and g(x) if h(x) = f(g(x))

  16. Closure • If , find f(x) and g(x) if h(x) = f(g(x)) HW: p.144 #91-101 odds 1.2 1.5 1.6 Quiz on Friday

  17. 1.2 1.5 1.6 ReviewThurs Sept 26 • Do Now • For each function h(x), identify the f(x) and g(x) such that h(x) = f(g(x))

  18. HW Review: p.144 #91-101

  19. 1.2 1.5 1.6 Review • 1.2 Domain of Functions • 1.5 Finding increasing/decreasing intervals • 1.5 Finding max/mins • 1.5 Piecewise Functions • 1.6 Function Composition • 1.6 Decomposing Compositions • Full period quiz – 50 pt quiz

  20. Closure • What is a composition? How do you evaluate one? • 1.2 1.5 1.6 quiz tomorrow!

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