To determine which function is undefined for [tex]\(x=0\)[/tex], let's analyze each function step-by-step.
1. Function 1: [tex]\( y = \sqrt[3]{x-2} \)[/tex]
- This is the cube root function.
- For [tex]\( x = 0 \)[/tex], it becomes:
[tex]\[
y = \sqrt[3]{0-2} = \sqrt[3]{-2}
\][/tex]
- Cube roots of negative numbers are defined. Therefore, [tex]\( y = \sqrt[3]{-2} \)[/tex] is defined.
2. Function 2: [tex]\( y = \sqrt{x-2} \)[/tex]
- This is the square root function.
- For [tex]\( x = 0 \)[/tex], it becomes:
[tex]\[
y = \sqrt{0-2} = \sqrt{-2}
\][/tex]
- Square roots of negative numbers are not real numbers and are undefined in the set of real numbers. Therefore, [tex]\( y = \sqrt{-2} \)[/tex] is undefined.
3. Function 3: [tex]\( y = \sqrt{x+2} \)[/tex]
- This is also a square root function.
- For [tex]\( x = 0 \)[/tex], it becomes:
[tex]\[
y = \sqrt{0+2} = \sqrt{2}
\][/tex]
- Square roots of positive numbers are defined. Therefore, [tex]\( y = \sqrt{2} \)[/tex] is defined.
In conclusion, the function that is undefined for [tex]\( x = 0 \)[/tex] is [tex]\( y = \sqrt{x-2} \)[/tex].
Thus, the correct answer is:
[tex]\[ \textbf{Option 2: } y = \sqrt{x-2} \][/tex]
