hw01-sol

hw01-sol - Homework # 1 Due: 05/23/06 #1 Find the domain of...

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Homework # 1 Due: 05/23/06 #1 Find the domain of the following functions a) f ( x ) = 3 x 2 - 2 x + 1 x 2 - 4 x - 21 The domains of 3 x 2 - 2 x + 1 and x 2 - 4 x - 21 are all real numbers. So the domain of f is all real numbers except where x 2 - 4 x - 21 = 0. So x 2 - 4 x - 21 = 0 ( x - 7)( x + 3) = 0 x = 7 , - 3 So the domain is all real numbers except x = - 3 and x = 7. b) f ( x ) = x + 3 - x - 2 + 3 x 2 - 1 You can only take the square root of a positive number, so the domain of x + 3 is where x + 3 0, or x ≥ - 3. Similarly, the domain of x - 2 is x - 2 0 or x 2. You can take the cube root of any number, so the domain of 3 x 2 - 1 is all real numbers. The domain of f is the intersection of all of these. So it’s all numbers that are bigger than both -3 and 2. This means the domain is x 2 or [2 , ) c) f ( x ) = 1 x 2 - 1 The domain of f is the domain of x 2 - 1 minus the points where it’s zero. The domain of x 2 - 1 is where x 2 - 1 0. To ±nd where this is positive, ±rst ±nd where it’s zero and then test values. x 2 - 1 = 0 ( x + 1)( x - 1) = 0 x = - 1 , 1 So now we look at the number line: - 1 1 1
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So we want to test values less than -1, between -1 and 1, and greater than 1. Plugging in
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hw01-sol - Homework # 1 Due: 05/23/06 #1 Find the domain of...

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