To determine which of the given functions is not defined for all values of [tex]\( x \)[/tex], we will analyze each function one by one:
1. Function [tex]\( f(x) = x^3 \)[/tex]
- This is a polynomial function, which is defined for all values of [tex]\( x \)[/tex] in the real number system. There are no restrictions on [tex]\( x \)[/tex] for a cubic function.
- Conclusion: This function is defined for all [tex]\( x \)[/tex].
2. Function [tex]\( f(x) = (-x)^{\frac{1}{2}} \)[/tex]
- Here we consider the expression [tex]\( (-x)^{\frac{1}{2}} \)[/tex], which is equivalent to the square root of [tex]\(-x\)[/tex].
- The square root function, [tex]\( \sqrt{\cdot} \)[/tex], is only defined for non-negative numbers in the real number system. Therefore, [tex]\( -x \)[/tex] must be non-negative for this function to be real, which implies [tex]\( x \leq 0 \)[/tex].
- Hence, [tex]\( f(x) = (-x)^{\frac{1}{2}} \)[/tex] is not defined for positive [tex]\( x \)[/tex].
- Conclusion: This function is not defined for all [tex]\( x \)[/tex].
3. Function [tex]\( f(x) = (-x)^{\frac{1}{3}} \)[/tex]
- The expression [tex]\( (-x)^{\frac{1}{3}} \)[/tex] is equivalent to the cube root of [tex]\(-x\)[/tex].
- The cube root function is defined for all real numbers, since taking the cube root of a negative number results in a real negative number.
- Conclusion: This function is defined for all [tex]\( x \)[/tex].
4. Function [tex]\( f(x) = 1 - \lvert x \rvert^{\frac{1}{2}} \)[/tex]
- The expression [tex]\( \lvert x \rvert^{\frac{1}{2}} \)[/tex] is the square root of the absolute value of [tex]\( x \)[/tex], which is defined for all real numbers because both [tex]\( x \)[/tex] and [tex]\( -x \)[/tex] are non-negative when taken as an absolute value.
- Since [tex]\( \lvert x \rvert^{\frac{1}{2}} \)[/tex] is always defined, the function [tex]\( f(x) = 1 - \lvert x \rvert^{\frac{1}{2}} \)[/tex] is also defined for all real numbers.
- Conclusion: This function is defined for all [tex]\( x \)[/tex].
5. Function [tex]\( f(x) = 1 - \lvert x \rvert^{\frac{1}{3}} \)[/tex]
- The expression [tex]\( \lvert x \rvert^{\frac{1}{3}} \)[/tex] is the cube root of the absolute value of [tex]\( x \)[/tex], which is also defined for all real numbers because the cube root function can handle both positive and negative inputs.
- Since [tex]\( \lvert x \rvert^{\frac{1}{3}} \)[/tex] is always defined, the function [tex]\( f(x) = 1 - \lvert x \rvert^{\frac{1}{3}} \)[/tex] is defined for all real numbers.
- Conclusion: This function is defined for all [tex]\( x \)[/tex].
After analyzing all the functions, the conclusion is that the function [tex]\(\boxed{2}\)[/tex] which is [tex]\( f(x) = (-x)^{\frac{1}{2}} \)[/tex], is not defined for all values of [tex]\( x \)[/tex]. It is only defined when [tex]\( x \leq 0 \)[/tex].
