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UNIT I: FUNCTIONS Combinations/Compositions 1.5

UNIT I: FUNCTIONS Combinations/Compositions 1.5. JMerrill, 2006 Revised 2008. Learning Goal. Day 1: To perform operations with functions and to determine the domains of the resulting functions. Operations: Sum. Let Find f(x) + g(x) . Operations: Difference. Let Find f(x) – g(x).

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UNIT I: FUNCTIONS Combinations/Compositions 1.5

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  1. UNIT I: FUNCTIONSCombinations/Compositions1.5 JMerrill, 2006 Revised 2008

  2. Learning Goal • Day 1: To perform operations with functions and to determine the domains of the resulting functions.

  3. Operations: Sum Let Find f(x) + g(x)

  4. Operations: Difference Let Find f(x) – g(x)

  5. Operations: Product Let Find

  6. Operations: Quotient Let Find

  7. You Do: Let Find:f(x) + g(x) f(x)•g(x) f(x) - g(x) g(x) - f(x)

  8. Summary of Notations

  9. Domains • The domain of an arithmetic combination of functions consists of all real numbers that are common to the domains of f and g. In the case of the quotient f(x)/g(x), there is a further restriction that g(x) ≠ 0

  10. You Do: Let Find the Domains:f(x) + g(x) f(x)•g(x) f(x) - g(x) g(x) - f(x)

  11. Graphs Example: f(-2) + g(-2) = 2 + 1 = 3 f(-1) + g(-1) 0 + 1 = 1 f(0) + g(0) -1 + 0 = -1 f(1) + g(1) -1 + 2 = 1 f(2) + g(2) 1 + 2 = 3

  12. Graphs

  13. Gateway Problem • If f(x) = 2x – 5 and g(x) = 1 - x • Find: • A) (f+g)(x) • B) (f•g)(x) x - 4 -2x2 + 7x - 5

  14. Learning Goal • To perform operations with functions and to determine the domains of the resulting functions. • Were you successful? That’s it for today! Homework time… 

  15. Learning Goal • Day 2: To perform compositions with functions and to determine the domains of the resulting functions.

  16. Composition of Functions • In the real-world, most things are not modeled by simple linear equations. Some are based on a system of equations (functions), others are based on a composition of functions. Sometimes a function’s output depends on some input that is itself an output of another function.

  17. Compositions Con’t • For example, the amount you pay on your income tax depends on the amount of adjusted gross income (on your Form 1040), which, in turn, depends on your annual earnings.

  18. Composition Example • In chemistry, the process to convert Fahrenheit temperatures to Kelvin units • This 2-step process that uses the output of the first function as the input of the second function. This formula gives the Celsius temp. that corresponds to the Fahrenheit temp. This formula converts the Celsius temp. to Kelvins

  19. Composition of Functions: A Graphing Approach

  20. f(g(0)) = g(f(0)) = (f°g)(3) = (f°g)(-3) = (g°f)(4) = (f°g)(4) = You Do 4 4 f(x) 3 3 g(x) 0 0

  21. Compositions: Algebraically • Given f(x) = 3x2 and g(x) = 5x+1 • Find f(g(2)) Find g(f(4)) • g(2)=5(2)+1 = 11 • f(11) = 3(11)2 • =363 How much is f(4)? g(48) = 5(48)+1=241

  22. Compositions: Algebraically Con’t • Given f(x) = 3x2 and g(x) = 5x+1 • Find f(g(x)) Find g(f(x)) • What does g(x)=? • f(5x+1) • =3(5x+1)2 • =3(25x2+10x+1) • =75x2+30x+3 What does f(x)=? g(3x2) = 5(3x2)+1=15x2+1

  23. You Do • f(x)=4x2-1 g(x) = 3x • Find:

  24. Decomposing Functions • If h(x) = (3x2 – 4x + 1)5. Find f(x) and g(x) such that f(g(x)) = h(x) • If f(x) = x5 and g(x) = 3x2 – 4x + 1, then f(g(x) = (3x2 – 4x + 1)5

  25. You Do • If h(x) = find f(x) and g(x) such that f(g(x)) = h(x)

  26. Gateway Problem • If f(x) = x2 and g(x) = x – 1, find: • A. (f o g)(x) • B. (g o f)(x)

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