To determine the correct description of the function \( y = 4 \sqrt[3]{\sqrt{x} - 2} \), let's analyze the function's domain and range.
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Domain: The expression under the cube root, \( \sqrt{x} - 2 \), needs to be defined for the function to make sense. This requires:
- \( \sqrt{x} \) is defined for \( x \geq 0 \).
- \( \sqrt{x} - 2 \) can be any real number, meaning there is no further restriction on \( x \) after ensuring \( x \geq 0 \).
- Therefore, the domain of the function is \( x \geq 0 \), which is the set of all non-negative real numbers.
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Range: The value of \( y \) can take on all real numbers because:
- As \( x \) approaches 0, \( \sqrt{x} - 2 \) approaches \( -2 \) and thus \( y \) approaches \( 4 \cdot (-2)^{1/3} \), which is a negative number.
- As \( x \) increases, \( \sqrt{x} - 2 \) becomes positive, and hence \( y \) takes on all values upwards as it continues to increase.
- Therefore, the range of the function is the set of all real numbers.
From this analysis, we can conclude:
- The domain of the function is the set of all non-negative real numbers (from \( x \geq 0 \)).
- The range of the function is the set of all real numbers.
Thus, the correct response is:
The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers.