Answer :
To determine the interval where the function [tex]\( g(x) = \sqrt[3]{x - 3} \)[/tex] is positive, let's follow these detailed steps:
1. Understand the Function: The function given is [tex]\( g(x) = \sqrt[3]{x - 3} \)[/tex], which is the cube root of [tex]\( x - 3 \)[/tex]. This function involves a cube root, which is defined for all real numbers.
2. Positive Function Condition: To identify when [tex]\( g(x) \)[/tex] is positive, we need to find when the expression inside the cube root, [tex]\( x - 3 \)[/tex], is positive.
3. Set Up the Inequality: To find when the cube root function is positive:
[tex]\[ \sqrt[3]{x - 3} > 0 \][/tex]
4. Solve the Inequality:
[tex]\[ x - 3 > 0 \][/tex]
[tex]\[ x > 3 \][/tex]
5. Conclusion: The function [tex]\( g(x) = \sqrt[3]{x - 3} \)[/tex] is positive when [tex]\( x \)[/tex] is greater than 3. This is expressed as the interval:
[tex]\[ (3, \infty) \][/tex]
So, the correct answer is:
[tex]\[ (3, \infty) \][/tex]
1. Understand the Function: The function given is [tex]\( g(x) = \sqrt[3]{x - 3} \)[/tex], which is the cube root of [tex]\( x - 3 \)[/tex]. This function involves a cube root, which is defined for all real numbers.
2. Positive Function Condition: To identify when [tex]\( g(x) \)[/tex] is positive, we need to find when the expression inside the cube root, [tex]\( x - 3 \)[/tex], is positive.
3. Set Up the Inequality: To find when the cube root function is positive:
[tex]\[ \sqrt[3]{x - 3} > 0 \][/tex]
4. Solve the Inequality:
[tex]\[ x - 3 > 0 \][/tex]
[tex]\[ x > 3 \][/tex]
5. Conclusion: The function [tex]\( g(x) = \sqrt[3]{x - 3} \)[/tex] is positive when [tex]\( x \)[/tex] is greater than 3. This is expressed as the interval:
[tex]\[ (3, \infty) \][/tex]
So, the correct answer is:
[tex]\[ (3, \infty) \][/tex]
