3.5 Operations of Functions Objectives: 1.Form sum, difference, product, and quotient functions, and find their domains. 2.Form composite functions and.

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Presentation transcript:

3.5 Operations of Functions Objectives: 1.Form sum, difference, product, and quotient functions, and find their domains. 2.Form composite functions and find their domains.

Sum and Difference Functions The domain of the sum or difference functions is the set of all real numbers that are in both the domain of f AND the domain of g.

Example #1 Sum & Difference Functions A.Write a rule for.

Example #1 Sum & Difference Functions B.Find the domain of each. Domain of f : Domain of g : Domain of f + g and f – g : The set of all x such that x = 1 or x ≤ −1

Product & Quotient Functions The domain of the product function fg is the set of all real numbers that are in both the domain of f AND the domain of g. The domain of the quotient function f/g is the same as the product so long as g(x) ≠ 0.

Example #2 Product & Quotient Functions A.Write a rule for.

Example #2 Product & Quotient Functions B.Find the domain of each. Domain of f : Domain of g : For the quotient function, g(x) ≠ 0. The domain of fg is the set of all x such that x ≥ 2 The domain of f/g is the set of all x such that x > 2, the only difference being that 2 is excluded from the domain of fg.

Products with Constant Functions

Example #3 Products with Constant Functions

Composite Functions

Example #4 Composite Functions A. B.

Example #4 Composite Functions C. D. OR

Example #5 Finding the Domain of Composite Functions A. B. C.State the domain of each composite function. First identify the domain of the inside function. Domain of f (g(x)) : Since the domain of g is all real numbers, then the only restriction is that of the composite. Domain of g(f (x)) : Since the domain of f has restrictions and the composite doesn’t, then it shares its domain with f, x ≥ 2.

Example #6 Writing a Function as a Composite A. Write h(x) as a composite function two different ways. B.

Example #7 Compositions with Absolute Value Functions Graph f, and the composite function Describe the relationship between the graphs of f and in terms of transformations. In the composite function, the graph of f was reflected over the y-axis.

Example #8 Application A cylindrical container is being filled with water. After t minutes, the height of the water in the container is inches. The volume V of the water in the container is given by. Express the volume as a function of time by finding, and compute the volume at t = 2 minutes.