y=sqrt((4)/(x))-sqrt(3x)

Define Domain Criteria: We need to determine the set of all possible x values for which the function y = sqrt((4)/(x)) - sqrt(3x) is defined. The domain of a function is the set of all possible inputs (x values) that will give real number outputs for the function.First Term Analysis: The first term of the function is sqrt((4)/(x)). The square root function is only defined for non-negative numbers. Therefore, (4)/(x) must be greater than or equal to 0. This implies that x must be greater than 0 because if x were negative, the fraction (4)/(x) would be negative, which is not allowed under a square root.Second Term Analysis: The second term of the function is -sqrt(3x). Similarly, the square root function here is only defined for non-negative numbers. Therefore, 3x must be greater than or equal to 0. This implies that x must be greater than or equal to 0.Combine Conditions: Combining the conditions from both terms, we find that x must be greater than 0 to satisfy the first term and greater than or equal to 0 to satisfy the second term. The common condition that satisfies both is that x must be strictly greater than 0.Consider Exclusion: However, we must also consider that x cannot be zero because the term (4)/(x) would involve division by zero, which is undefined. Therefore, the domain of the function excludes x = 0.Final Domain: The domain of the function y = sqrt((4)/(x)) - sqrt(3x) is therefore all positive real numbers except zero. This can be expressed in set notation as {x | x > 0}.

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$y=\sqrt{\frac{4}{x}}-\sqrt{3x}$

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Algebra 1

Domain and range of square root functions: equations

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y=\sqrt{\frac{4}{x}}-\sqrt{3x}

Define Domain Criteria: We need to determine the set of all possible $x$ values for which the function $y = \sqrt{\left(\frac{4}{x}\right)} - \sqrt{3x}$ is defined. The domain of a function is the set of all possible inputs ( $x$ values) that will give real number outputs for the function.
First Term Analysis: The first term of the function is $\sqrt{\left(\frac{4}{x}\right)}$ . The square root function is only defined for non-negative numbers. Therefore, $\frac{4}{x}$ must be greater than or equal to $0$ . This implies that $x$ must be greater than $0$ because if $x$ were negative, the fraction $\frac{4}{x}$ would be negative, which is not allowed under a square root.
Second Term Analysis: The second term of the function is $-\sqrt{3x}$ . Similarly, the square root function here is only defined for non-negative numbers. Therefore, $3x$ must be greater than or equal to $0$ . This implies that $x$ must be greater than or equal to $0$ .
Combine Conditions: Combining the conditions from both terms, we find that $x$ must be greater than $0$ to satisfy the first term and greater than or equal to $0$ to satisfy the second term. The common condition that satisfies both is that $x$ must be strictly greater than $0$ .
Consider Exclusion: However, we must also consider that $x$ cannot be zero because the term $\frac{4}{x}$ would involve division by zero, which is undefined. Therefore, the domain of the function excludes $x = 0$ .
Final Domain: The domain of the function $y = \sqrt{\left(\frac{4}{x}\right)} - \sqrt{3x}$ is therefore all positive real numbers except zero. This can be expressed in set notation as \{x | x > 0\}.