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Section 1.5 Combinations of Functions

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Combinations of Functions Arithmetic Combinations of Functions The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the Case of f(x)/g(x), there is the further restriction that g(x) 0. Sum, Difference, Product and Quotient Functions 1. Sum 2. Difference 3. Product 4. Quotient

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Combinations of Functions Given Find: Try these: 1. (f + g)(x) 2. (f – g)(x) 3. (f g)(x) 4. (f / g)(x)

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Combinations of Funcitons Try these: Given: Find: 1. (f + g)(3) 2. (f – g)(-1) 3. (f g)(-2) 4. (g / f)(0) = 8 = - 8 = - 35 = -1/5

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Combinations of Functions Definition of Composition of Two Functions The composition of the function f with the function g is The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain of f.

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Combinations of Functions How to from the composite of two functions: Given: Step 1: Think of f(x) as being the primary function and g(x) being the secondary function. To form the composite substitute the secondary function into the primary function. Step 2: Simplify the result. Definition of composite Substitution of 4 – x 2 for x in x + 2

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Combinations of Functions How to from the composite of two functions (cont.): Given: Step 1: Think of g(x) as being the primary function and f(x) being the secondary function. To form the composite substitute the secondary function into the primary function. Step 2: Simplify the result. Definition of composite Substitution of x + 2 for x in 4 – x 2

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Combinations of Functions How to find the numeric value for the combination of two functions Given: Step 1: Find g(2). Step 2: Take the value for g(2) and substitute it into f(x). Step 3: The answer found in step 2 is the numeric value for

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Combinations of Functions How to find numeric values of composite graphically. Given: Step 1: Graph both functions on individual graphs. Step 2: Find g(0) f(x) = x g(x) = 2x + 1 Step 3: Take the value from Step 2 and substitute It in for x in f(x). Find its value on the graph. g(0) = 1 f(1) = -2

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Combinations of Functions Try these

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Combinations of Functions What you should have learned: 1. To add, subtract, multiply, and divide functions 2.To find the composition of one function with another function.

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