exam1_2009_sol

# B 3 points what is the domain of f solution f x is

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(b) [3 points] What is the domain of f ? Solution: f ( x ) is defined as long as 2 x 1 / 3 - 1 6 = 0. But if 2 x 1 / 3 - 1 = 0, then x 1 / 3 = 1 / 2 and so x = ( x 1 / 3 ) 3 = (1 / 2) 3 = 1 / 8. Thus the domain of f is { x : x 6 = 1 / 8 } , or in interval notation, ( -∞ , 1 / 8) (1 / 8 , ). (c) [3 points] What is the range of f ? Solution: The range of f is the same as the domain of f - 1 . But f - 1 ( x ) is defined as long as 2 x - 1 6 = 0. Thus the domain of f is { x : x 6 = 1 / 2 } , or in interval notation, ( -∞ , 1 / 2) (1 / 2 , ). 6. Let p ( x ) = x 4 + 2 x 3 - 11 x - 10 and q ( x ) = x 2 + 3 x + 5. (a) [8 points] Divide p by q . Solution: Performing the division: x 2 - x - 2 x 2 + 3 x + 5 ) x 4 + 2 x 3 - 11 x - 10 - x 4 - 3 x 3 - 5 x 2 - x 3 - 5 x 2 - 11 x x 3 + 3 x 2 + 5 x - 2 x 2 - 6 x - 10 2 x 2 + 6 x + 10 0 So x 4 + 2 x 3 - 11 x - 10 x 2 + 3 x + 5 = x 2 - x - 2. (b) [4 points] Factor p completely. Solution: Note that x 2 +3 x +5 has no real roots because 3 2 - 4 · 5 = - 11 < 0. So x 2 + 3 x + 5 has no linear factors, meaning that we cannot factor it further. On the other hand, x 2 - x - 2 = ( x - 2)( x + 1). Therefore, p ( x ) = ( x 2 + 3 x + 5)( x - 2)( x + 1), and it cannot be factored further. (c) [1 point] What are the zeros of p ? Page 3

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Math 171: Exam 1 Solution: From part (b), the only (real) numbers that make p ( x ) zero are x = 2 , - 1. 7. [10 points] Solve | x - 3 | = 1 - 4 x . ( Hint : You may want to break it into two cases.) Solution: We can break the problem into two cases. Case 1: If x - 3 0, then | x - 3 | = x - 3, so the equation becomes x - 3 = 1 - 4 x .
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