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Chapter 1: Problem 89

The domain of the function \(\mathrm{f}(\mathrm{x})=[1 / \sqrt{(|\mathrm{x}|-\mathrm{x})]}\) is (a) \((-\infty, \infty)\) (b) \((0, \infty)\) (c) \((-\infty, 0)\) (d) \((-\infty, \infty)-\\{0\\}\)

Short Answer

Expert verified
The domain of the function \(f(x) = \cfrac{1}{\sqrt{|x| - x}}\) is (c) \((- \infty, 0)\), as the function is defined for all negative values of x.

Step by step solution

01

Identify restrictions

In the given function, \(f(x) = \cfrac{1}{\sqrt{|x| - x}}\), the restrictions we need to consider are: 1. The denominator of a fraction must not be equal to zero. 2. The number inside the square root should be non-negative. Now we will analyze the expressions that could cause these restrictions. The denominator of the function is \(\sqrt{|x|-x}\) and we need to make sure it's not equal to 0.
02

Identify possible values of the argument of the square root

We have the argument of the square root: \(|x|-x\). Let's analyze it for different values of x. 1. If \(x\) is positive, \(|x|=x\) and the argument will be: \[x-x=0\] 2. If \(x\) is negative, \(|x|=-x\) and the argument will be: \[-x-x=-2x\] 3. If \(x\) is zero, \(|0|=0\) and the argument will be: \[0-0=0\]
03

Check the restrictions for the argument of the square root

Now let's check the restrictions: 1. For the positive values of \(x\), the argument will be 0. Therefore, the function is not defined for positive values of \(x\). 2. For the negative values of \(x\), the argument will be \(-2x\), which is always non-negative and not equal to 0. So, the function is defined for negative values of \(x\). 3. For \(x=0\), the argument will be 0. Therefore, the function is not defined for \(x=0\). Based on our analysis, the domain of the function \(f(x) = \cfrac{1}{\sqrt{|x| - x}}\) consists of all negative values of x. The correct answer is (c) \((- \infty, 0)\).

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