To analyze the function \( y = 4 \sqrt[3]{\sqrt{x} - 2} \), we need to determine the domain and range.
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Domain:
- The function involves a square root, which indicates that the expression inside it, \(\sqrt{x}\), must be non-negative. Therefore, \( x \) must be greater than or equal to 0.
- However, since we need \(\sqrt{x} - 2\) to be defined in terms of the cube root, and since cube roots are defined for all real numbers, there will be a restriction where \(\sqrt{x} \geq 2\) (or \(x \geq 4\)) to get non-negative values under the cube root.
- Thus, the domain is \( [4, \infty) \).
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Range:
- The smallest value of \( y \) when \( x = 4 \) is \( 4 \sqrt[3]{0} = 0 \).
- As \( x \) increases, \(\sqrt{x} - 2\) will also increase indefinitely, so \( y \) can rise without bound.
- Therefore, the range of the function is \( [0, \infty) \).
Given these findings, the correct description of the graph corresponds to the option stating that:
The domain of the function is the set of all non-negative real numbers and the range is the set of all non-negative real numbers.
However, since we found \( x \) must be greater than or equal to 4 for the function to be valid, and the range starts at 0, the precise match is:
The domain is the set of all real numbers greater than or equal to 4 and the range is the set of all non-negative real numbers.
If this precise option is not phrased as a response, select the closest accurate option.
1 answer