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 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 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 real numbers.

Answer 1

To analyze the function \( y = 4 \sqrt[3]{\sqrt{x} - 2} \):

1. Domain:

   • The function involves a square root, \( \sqrt{x} \), which means \( x \) must be non-negative for \( \sqrt{x} \) to be defined. Therefore, the domain is \( x \geq 0 \), which corresponds to the set of all non-negative real numbers.

2. Range:

   • The expression \( \sqrt{x} - 2 \) can take any real number based on the value of \( x \). Since \( \sqrt{x} \) starts at 0 (when \( x = 0 \)) and increases without bound as \( x \) increases, this means \( \sqrt{x} - 2 \) can go from \( -2 \) to \( +\infty \). When inputted into the cube root function and then multiplied by 4, the output \( y \) can also take all real numbers.

So, the resulting conclusions are:

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

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.