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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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MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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