To determine which function is undefined for [tex]\( x=0 \)[/tex], we need to analyze each function individually.
1. For [tex]\( y = \sqrt[3]{x-2} \)[/tex]:
- This is the cube root function of [tex]\( x-2 \)[/tex].
- The cube root function is defined for all real numbers.
- Substituting [tex]\( x=0 \)[/tex]:
[tex]\[
y = \sqrt[3]{0-2} = \sqrt[3]{-2}
\][/tex]
The cube root of [tex]\(-2\)[/tex] is defined.
Result: Defined
2. For [tex]\( y = \sqrt{x-2} \)[/tex]:
- This is the square root function of [tex]\( x-2 \)[/tex].
- The square root function is defined only for non-negative arguments.
- Substituting [tex]\( x=0 \)[/tex]:
[tex]\[
y = \sqrt{0-2} = \sqrt{-2}
\][/tex]
The square root of a negative number is not defined in the set of real numbers; it results in a complex number.
Result: Undefined (for real numbers)
3. For [tex]\( y = \sqrt[3]{x+2} \)[/tex]:
- This is the cube root function of [tex]\( x+2 \)[/tex].
- The cube root function is defined for all real numbers.
- Substituting [tex]\( x=0 \)[/tex]:
[tex]\[
y = \sqrt[3]{0+2} = \sqrt[3]{2}
\][/tex]
The cube root of 2 is defined.
Result: Defined
4. For [tex]\( y = \sqrt{x+2} \)[/tex]:
- This is the square root function of [tex]\( x+2 \)[/tex].
- The square root function is defined only for non-negative arguments.
- Substituting [tex]\( x=0 \)[/tex]:
[tex]\[
y = \sqrt{0+2} = \sqrt{2}
\][/tex]
The square root of 2 is defined.
Result: Defined
Therefore, the function that is undefined for [tex]\( x=0 \)[/tex] is:
[tex]\[ y = \sqrt{x-2} \][/tex]
