# HW3 Sol - Math 2312 Homework 3 Summer 2010 Due Wednesday at...

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Math 2312 Homework 3 Summer 2010 Due: Wednesday, June 16, 2010 at the beginning of class 1. Find the zeros of the function a) ) 1 3 ( ) 4 3 )( 3 2 ( ) ( x x x x x f Solution: 2 , 2 ) 2 )( 2 ( 3 ) 4 ( 3 12 3 3 12 6 ) 1 3 ( ) 4 3 )( 3 2 ( ) ( 2 2 2 2 x x x x x x x x x x x x x x x f b) 6 5 6 ) ( 2 2 x x x x x f Solution: ) 3 )( 2 ( ) 6 ( 6 5 6 ) ( 2 2 x x x x x x x x x f Since only the numerator is zero at x = 0 and x = 6 (not the denominator), then the zeros are x = 0 and x = 6. 2. Find the domain of f, the zeros of f , the interval(s) where 0 ) ( x f , and the interval(s) where 0 ) ( x f for the function, 1 2 1 ) ( x x x x f

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Hint: first, rewrite f(x) as one fraction. Solution:   1 2 2 1 2 2 2 1 2 ) 1 ( 2 1 2 1 2 ) 1 ( 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 ) ( 2 x x x x x x x x x x x x x x x x x x x x x x x x x f Domain: 0 1 x To “avoid” an even root of a negative number and to “avoid” division by zero, so } 1 : { : x x D or ) , 1 ( Zeros: Where only the numerator is 0: at x = 2 Sign analysis: By definition, the square root function is always nonnegative if the radicand is nonnegative, so 0 1 x for x>1. so only the numerator affects the sign: if x>2 then f(x)>0 and if x<2, then f(x)<0: f(x)>0 on ) , 2 ( and f(x)<0 on ) 2 , (  but since x cannot be less than or equal to one,
then we must say that f(x)<0 on ) 2 , 1 ( only. 3.

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HW3 Sol - Math 2312 Homework 3 Summer 2010 Due Wednesday at...

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