To determine which function is undefined at \( x = 0 \), let's analyze each function one by one:
1. Function: \( y = \sqrt[3]{x - 2} \)
- Substitute \( x = 0 \):
[tex]\[
y = \sqrt[3]{0 - 2} = \sqrt[3]{-2}
\][/tex]
- The cube root of \(-2\) is defined for all real numbers.
- Therefore, this function is defined at \( x = 0 \).
2. Function: \( y = \sqrt{x - 2} \)
- Substitute \( x = 0 \):
[tex]\[
y = \sqrt{0 - 2} = \sqrt{-2}
\][/tex]
- The square root of a negative number is not defined in the set of real numbers.
- Therefore, this function is undefined at \( x = 0 \).
3. Function: \( y = \sqrt[3]{x + 2} \)
- Substitute \( x = 0 \):
[tex]\[
y = \sqrt[3]{0 + 2} = \sqrt[3]{2}
\][/tex]
- The cube root of 2 is defined for all real numbers.
- Therefore, this function is defined at \( x = 0 \).
4. Function: \( y = \sqrt{x + 2} \)
- Substitute \( x = 0 \):
[tex]\[
y = \sqrt{0 + 2} = \sqrt{2}
\][/tex]
- The square root of 2 is defined in the set of real numbers.
- Therefore, this function is defined at \( x = 0 \).
By analyzing all four functions, we can conclude that the function [tex]\( y = \sqrt{x - 2} \)[/tex] is the only one that is undefined at [tex]\( x = 0 \)[/tex].
