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2.3 Combinations of Functions Introductory MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences (12 th Edition) Copyright © 2009Carol A. Marinas, Ph.D.

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Combining Functions If f(x) = 3x + 1 and g(x) = x 2 + 5x, find the following: (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) (fg)(x) = f(x) g(x) f (x) = f (x) for g(x) ≠ 0 g g (x) (f + g)(x) = f(x) + g(x) (f + g)(x) = (3x + 1) + (x 2 + 5x) (f + g)(x) = x 2 + 8x + 1 (f – g)(x) = f(x) – g(x) (f – g)(x) = (3x + 1) – (x 2 + 5x) (f – g)(x) = – x 2 – 2x + 1 (fg)(x) = f(x) g(x) (fg)(x) = (3x + 1) (x 2 + 5x) (fg)(x) = 3x 3 + 16x 2 + 5x f (x) = f (x) for g(x) ≠ 0 g g (x) f (x) = 3x + 1 for x ≠ 0, – 5 g x 2 + 5x Copyright © 2009Carol A. Marinas, Ph.D.

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Composition of Functions The composition combines two functions by applying one function to a number and then applying the other function to the result. Domain of g Range of g Domain of f Range of f X f ₀ g fg g(x)g(x) f (g(x)) = (f ₀ g)(x) Copyright © 2009Carol A. Marinas, Ph.D.

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Composition of Functions The function h(x) = (3x + 1) 2 is the composition combines two functions. The first function is 3x + 1 and the second function is to square the result. So g(x) = 3x + 1 and f(x) = x 2. X h(x) = (f ₀ g)(x) = (3x + 1) 2 fg g(x) = 3x + 1 (f ₀ g)(x) = f (g(x)) = f (3x + 1) = (3x + 1) 2 Copyright © 2009Carol A. Marinas, Ph.D.

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Composition of Functions With the function h(x) = (3x + 1) 2, find h(5) using the two-step composition method. 5 h(5) = (f ₀ g)(5) = (16) 2 = 256 fg g(x) = 35 + 1 = 16 (f ₀ g)(x) = f (g(x)) = f (16) = (16) 2 h(5) = 256 Copyright © 2009Carol A. Marinas, Ph.D.

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Review of Section 2.3 If f(x) = 1 – x and g(x) = 2x 2, find the following: 1.(f + g ) (3) 2.(g – f) (x) 3.(fg)(1) 4.(gf)(x) 5.g (x) [state the domain of the answer] f 6.f(3) 7.g(– 2) 8.(g ⁰ f )(3) 9.(g ⁰ f )(x) 10.(f ⁰ f )(x) Click mouse to check your answers. ANSWERS 1.16 2.2x 2 + x – 1 3.0 4.– 2x 3 + 2x 2 5. 2 x 2 [All Reals except 1] 1 – x 6.– 2 7.8 8.8 9.2(1 – x ) 2 10. x Copyright © 2009Carol A. Marinas, Ph.D.

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Next: Inverse Functions The compositions of functions will be used to prove that two functions are inverses. Copyright © 2009Carol A. Marinas, Ph.D.

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Warm-up Arithmetic Combinations (f+g)(x) = f(x) + g(x) (f-g)(x) = f(x) – g(x) (fg)(x) = f(x) ∙ g(x) (f/g)(x) = f(x) ; g(x) ≠0 g(x) The domain for these.

Warm-up Arithmetic Combinations (f+g)(x) = f(x) + g(x) (f-g)(x) = f(x) – g(x) (fg)(x) = f(x) ∙ g(x) (f/g)(x) = f(x) ; g(x) ≠0 g(x) The domain for these.

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