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What is the domain of the function $$u(x) = \frac{1}{\sqrt{|x|}}?$$ Express your answer in interval notation.

 Aug 8, 2023
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The domain of a function is defined as the set of inputs that yields a valid output.

 

After scanning the function definition, I spotted a square root within the definition of u(x). Only nonnegative inputs are allowed for square roots. However, within the square root, there is an absolute value sign. The absolute value sign takes the input and returns the distance from 0 on the number line. Distances are nonnegative, so there is no input restriction on the square root portion of the function definition.

 

I also spot a fraction. The restriction for fractions is that the denominator must be nonzero. If we consider x = 0 as an input, then the denominator will be 0, which is not allowed. Therefore, 0 is not within the domain of u(x). The input x = 0 is the only example of a real number that fails to produce a valid output.

 

In interval notation, this domain is represented like this:\((- \infty,0) \cup (0, \infty)\).

 Aug 9, 2023

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