Stanley ocken m19500 precalculus chapter 26 combining

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Stanley Ocken M19500 Precalculus Chapter 2.6: Combining functions
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Combining and composing functions Example 6: Given f ( x ) = 3 x - x 2 and g ( x ) = 4 x + 7 , find f ( g ( x )) and g ( f ( x ) and rewrite each as a simplified product. Solution: f ( g ( x )) = f (4 x + 7) = 3(4 x + 7) - (4 x + 7) 2 = (4 x + 7)(3 - (4 x + 7)) = (4 x + 7)(3 - 4 x - 7) = (4 x + 7)( - 4 x - 4) = - 4( x + 1)(4 x + 7) g ( f ( x )) = 4(3 x - x 2 ) + 7 = - 4 x 2 + 12 x + 7 . To decide whether this factors, calculate the discriminant D = b 2 - 4 ac = 144 - 4( - 4)(7) = 144 - 112 = 32 . Since D = 32 = 4 2 is not a whole number, the quadratic factoring criterion from Section 1.5 tells us that g ( f ( x )) = - 4 x 2 + 12 x + 7 does not factor and is therefore a simplified product. Note that g ( f ( x )) and f ( g ( x )) are unrelated: in general, composition is not commutative. Stanley Ocken M19500 Precalculus Chapter 2.6: Combining functions
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Combining and composing functions Example 7: Given f ( x ) = 3 x + 2 5 and g ( x ) = 5 x - 2 3 , find f ( g ( x )) and g ( f ( x )) and rewrite each as a simplified sum. Solution: f ( g ( x )) = f ( 5 x - 2 3 ) = 3 ( 5 x - 2 3 ) + 2 5 = 5 x - 2 + 2 5 = 5 x 5 = x g ( f ( x )) = g ( 3 x +2 5 ) = 5 ( 3 x +2 5 ) - 2 4 = 3 x + 2 - 2 3 = 3 x 3 = x Note that g ( f ( x )) and f ( g ( x )) are both equal to x . If you start with any number, apply either function, then apply the other function, you get back to the original number. We say that the functions f and g undo each other. The more technical language is that they are inverse functions, which we will study in detail in Chapter 2.7. The problem on the next slide is a warmup for the material in that chapter. Stanley Ocken M19500 Precalculus Chapter 2.6: Combining functions
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Combining and composing functions Example 8: Solve y = 3 x - 2 5 x +7 for x and check your answer. Solution: The given equation is y = 3 x - 2 5 x +7 Multiply both sides by 5 x + 7 (5 x + 7) y = 3 x - 2 Remove parentheses 5 xy + 7 y = 3 x - 2 Terms with x to left side 5 xy - 3 x + 7 y = - 2 Terms with y on right side 5 xy - 3 x = - 2 - 7 y Factor out x on left side x (5 y - 3) = - 2 - 7 y Divide by 5 y - 3 x = - 2 - 7 y 5 y - 3 Multiply answer by - 1 - 1 x = 7 y +2 - 5 y +3 The check is on the following slide.
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