Use the image to answer the question.

Question

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 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 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 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.

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

Answer 1

To analyze the function \( y = 4 \sqrt[3]{\sqrt{x} - 2} \), we can determine the domain and range.

1. Domain: The expression inside the cube root operator \(\sqrt{x} - 2\) must be defined. The square root function \(\sqrt{x}\) is defined for \(x \geq 0\). This means the domain starts from 0 and goes to infinity, so the domain is all non-negative real numbers: \( [0, \infty) \).

2. Range: Since the function involves a cube root, which can take any real number value, and then it is multiplied by 4 (a positive constant), the function can produce both positive and negative values. Therefore, the range encompasses all real numbers: \( (-\infty, \infty) \).

Based on the analysis above, the correct description that corresponds to the graphed function 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.