Identify the domain and range of the following radical function. \( y=\sqrt[3]{x-4} \)

To determine the **domain** and **range** of the radical function \( y = \sqrt[3]{x - 4} \), let's analyze the function step by step. ### Understanding the Function The given function is a **cube root function**, which is a type of radical function. The general form of a cube root function is: \[ y = \sqrt[3]{x - h} + k \] where: - \( h \) is the horizontal shift. - \( k \) is the vertical shift. In your function: - \( h = 4 \) (horizontal shift to the right by 4 units) - There is no vertical shift, so \( k = 0 \) ### Domain of \( y = \sqrt[3]{x - 4} \) **Domain** refers to all possible input values (x-values) for which the function is defined. - **Cube Root Characteristics**: The cube root function is defined for all real numbers because you can take the cube root of any real number (positive, negative, or zero). \[ \sqrt[3]{x - 4} \text{ is defined for all } x \in \mathbb{R} \] **Therefore, the domain is:** \[ \boxed{\text{Domain: } \mathbb{R} \text{ (all real numbers)}} \] ### Range of \( y = \sqrt[3]{x - 4} \) **Range** refers to all possible output values (y-values) the function can produce. - **Cube Root Characteristics**: Similarly, the cube root function can produce any real number as output because as \( x \) increases or decreases without bound, \( y \) also increases or decreases without bound. \[ y = \sqrt[3]{x - 4} \text{ can produce any } y \in \mathbb{R} \] **Therefore, the range is:** \[ \boxed{\text{Range: } \mathbb{R} \text{ (all real numbers)}} \] ### Summary - **Domain**: All real numbers \( (\mathbb{R}) \) - **Range**: All real numbers \( (\mathbb{R}) \) So, the function \( y = \sqrt[3]{x - 4} \) is defined for every real number \( x \), and it can produce every real number \( y \) as an output.