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A curve labeled y equals 4 root index 3 start root x minus 2 end root is plotted on a coordinate plane. The curve begins in quadrant 3 sloping upward from left to right, briefly passes through quadrant 4, and rises through quadrant 1.

Select the description that corresponds to the graphed function.

(1 point)
Responses

The domain and range of the function are both the set of all real numbers.
The domain and range of the function are both the set of all real numbers.

The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers.
The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers.

The domain of the function is the set of all real numbers and the range is the set of all non-negative real numbers.
The domain of the function is the set of all real numbers and the range is the set of all non-negative real numbers.

The domain and range of the function are both the set of all non-negative real numbers.
The domain and range of the function are both the set of all non-negative real numbers.
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1 answer

To determine the correct description of the function \( y = 4\sqrt[3]{x - 2} \), we need to analyze its domain and range.

  1. Domain:

    • The function \( y = 4\sqrt[3]{x - 2} \) involves a cube root, which is defined for all real numbers. Therefore, there are no restrictions on \( x \). The domain is all real numbers, \( (-\infty, \infty) \).

  2. Range:

    • The cube root function can produce all real outputs as \( x \) varies across all real numbers. Therefore, the range is also all real numbers, \( (-\infty, \infty) \).

Based on this analysis, the correct response is:

The domain and range of the function are both the set of all real numbers.