Question 2 50 pts Identify the domain and range of the following radical function. \[ y=\sqrt[3]{x}-3 \] Domain: \( x \geq 0 \) Range: \( y \geq 1 \) Domain: \( x \geq 1 \) Range: \( y \geq 3 \) Domain: All Real Numbers Range: All Real Numbers Domain: \( x \geq 3 \) Range: \( y \geq-1 \)

To determine the **domain** and **range** of the function \( y = \sqrt[3]{x} - 3 \), let's analyze the function step by step. ### 1. Domain The **domain** of a function consists of all possible input values (\( x \)) for which the function is defined. - The cube root function, \( \sqrt[3]{x} \), is defined for **all real numbers**. Unlike even roots (like square roots) which are only defined for non-negative numbers, cube roots can take any real number as input. - Subtracting 3 from \( \sqrt[3]{x} \) does not impose any additional restrictions on the domain. **Therefore, the domain is all real numbers.** ### 2. Range The **range** of a function consists of all possible output values (\( y \)) that the function can produce. - Since \( \sqrt[3]{x} \) can produce any real number, subtracting 3 from it simply shifts the output values down by 3 units. - This shift does not limit the range; \( y \) can still take **any real value**. **Therefore, the range is all real numbers.** ### Conclusion The correct identification is: - **Domain:** All Real Numbers - **Range:** All Real Numbers **Answer:** Domain: All Real Numbers Range: All Real Numbers