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5.1 Combining Functions

5.1 Combining Functions. Perform arithmetic operations on functions Review function notation Perform composition of functions. Operations on Functions. If f ( x ) and g ( x ) both exist, the sum, difference, product, quotient and composition of two functions f and g are defined by.

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5.1 Combining Functions

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  1. 5.1 Combining Functions Perform arithmetic operations on functions Review function notation Perform composition of functions

  2. Operations on Functions If f(x) and g(x) both exist, the sum, difference, product, quotient and composition of two functions f and g are defined by

  3. Operations on Functions and Domains The domains of the sum, difference, and product of fand g include x-values that are in both the domain of fand the domain of g.

  4. Example: Evaluating combinations of functions (1 of 3) If possible, use the graph representation of f and g to evaluate (f + g)(4), (f− g)(−2), (fg)(1), and

  5. Example: Evaluating combinations of functions (2 of 3) Solution f(4) = 9, g(4) = 2 (f + g)(4) = 9 + 2 = 11 f(−2) = −3, g(−2) is undefined, so (f+ g) is undefined

  6. Example: Evaluating combinations of functions (3 of 3) f(1) = 3, g(1) = 1 (fg)(3) = 3(1) = 3

  7. Example: Performing arithmetic operations on functions symbolically (1 of 6)

  8. Example: Performing arithmetic operations on functions symbolically (2 of 6)

  9. Example: Performing arithmetic operations on functions symbolically (3 of 6)

  10. Example: Performing arithmetic operations on functions symbolically (4 of 6) b. (fg)(0) is not defined, since 0 is not in the domain of f(x).

  11. Example: Performing arithmetic operations on functions symbolically (5 of 6)

  12. Example: Performing arithmetic operations on functions symbolically (6 of 6)

  13. Example: Evaluating function notation (1 of 2) Let g(x) = 3x² − 6x + 2. Evaluate each expression. a. g(2) b. g(k) c. g(x²) d. g(x + 2) Solution a. g(2) = 3(2)² − 6(2) + 2 = 12 − 12 + 2 = 2 b. g(k) = 3k² − 6k + 2

  14. Example: Evaluating function notation (2 of 2) d. g(x + 2) = 3(x + 2)² − 6(x + 2) + 2 = 3(x² + 4x + 4) − 6(x + 2) + 2 = 3x² + 12x + 12 − 6x− 12 + 2 = 3x² + 6x + 2

  15. Composition of Functions (1 of 2)

  16. Domain of Composition of Functions

  17. Composition of Functions (2 of 2)

  18. Example: Evaluating a composite function symbolically (1 of 5)

  19. Example: Evaluating a composite function symbolically (2 of 5)

  20. Example: Evaluating a composite function symbolically (3 of 5)

  21. Example: Evaluating a composite function symbolically (4 of 5)

  22. Example: Evaluating a composite function symbolically (5 of 5)

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