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Duplex Fractions, f(x), and composite functions

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[f(x) = Find f -1 (x)] A.[3x – 5 ] B.[3x – 15 ] C.[1.5x – 7.5 ] D.[Option 4]

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[h(x)= x 3 - 5. Find h -1 (x) ] A.[Option 1] B.[x 3 + 5 ] C.[Option 3] D.[Option 4]

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[Simplify] 25.0 0.1

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[Simplify] 1 0.1

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Which are wrong? A)-3 2 = -9 B)(-3) 2 = 9 C) x 2 when x = -3 is -9 D) 2(4) 2 = 64 E) 5 – (-2) 2 = 9 (-3) 2 = 9 2(16) = 32 5 - 4 = 1

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f(x) It means to simplify when x = ( ) Ex: f(x) = 3x + 1. Find f(-2) means 3(-2) + 1 = -5 Find f(a) means 3(a) + 1 = 3a + 1 Find f(a + 3) means 3(a+3) + 1 = 3a + 10 Or find y when x is ( ) f(1) is when x is 1 so 2 f(-4) is -2

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[Find f(3)] A.[3] B.[-1] C.[0] D.[2]

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[Find the value of f(4) and g(-10) if f(x)=-8x-8 and g(x)=2x 2 -22x] A.[-24, -2208] B.[-40, 420] C.[80, 8] D.[-16, 102]

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©1999 by Design Science, Inc.13 Composition of functions Composition of functions is the successive application of the functions in a specific order. Given two functions f and g, the composite function is defined by and is read “f of g of x.” The domain of is the set of elements x in the domain of g such that g(x) is in the domain of f. – Another way to say that is to say that “the range of function g must be in the domain of function f.”

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©1999 by Design Science, Inc.14 A composite function x g(x)g(x) f(g(x)) domain of g range of f range of g domain of f g f

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©1999 by Design Science, Inc.15 A different way to look at it… Function Machine x Function Machine g f

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f(x) = 3x + 2 g(x) = 2x - 5 f o g(3) f(g(3)) = f(2(3) – 5) = f(1) = 3(1) + 2 = 5 Plug 3 into g, get the answer, give it to f g o f(3) g(f(3)) = g(3(3) + 2) = g(11) = 2(11) – 5 = 17 Plug 3 into f, get the answer, give it to g

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f(x) = 3x 2 - 1 g(x) = x - 5 Find f o g (-2) and g o f(-2) f o g (-2) = f(g(-2)) = f(-2-5) = f(-7) = 3(-7) 2 – 1 = 146 go f (-2) = g(f(-2)) = g(3(-2) 2 - 1) = g(11)= 6

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f(x) = 3x + 2 g(x) = 2x - 5 f o g(x) f(g(x)) = f(2x – 5) = 3(2x-5)+ 2 = 6x-15+2 = 6x-13 Plug x into g, get equation, give it to f g o f(x) g(f(x)) = g(3x+2) = 2(3x+2)-5 = 6x +4-5= 6x-1 Plug x into f, get equation, give it to g

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Two functions, f(x) and g(x), are inverses if and only if fog(x)=x and g o f(x)=x Ex: f(x) = 3x + 2 g(x)= f o g(x) = f( ) = 3 + 2 = x – 2 + 2 = x g o f(x) = = 3x/3 = x

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Are the following functions inverses? Answer: Yes!

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Function A function is a mathematical “rule” that for each “input” (x-value) there is one and only one “output” (y – value). A function has a domain (input or x) and a range (output or y) A one-to-one function has only one x for each y!

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Examples of a Function { (2,3) (4,6) (7,8)(-1,2)(0,4)} 4 -2 1 8 -4 2

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4 -2 1 8 -4 2 Non – Examples of a Function {(1,2) (1,3) (1,4) (2,3)} Vertical Line Test – if it passes through the graph more than once then it is NOT a function.

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You Do: Is it a Function? 1.{(2,3) (2,4) (3,5) (4,1)} 2.{(1,2) (-1,3) (5,3) (-2,4)} 3. 4. 5. 0 -3 4 1 -5 9

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We need a common denominator to add these fractions.

We need a common denominator to add these fractions.

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