Questions: Which of the following is not a function for all values of x? f(x) = x^3 f(x) = (-x)^(1/2) f(x) = (-x)^(1/3) f(x) = 1 - x^(1/2) f(x) = 1 - x^(1/3)

Solution Steps

To determine which of the given expressions is not a function for all values of \( x \), we need to analyze the domain of each function. A function is not defined for all \( x \) if there are values of \( x \) for which the expression is undefined. Specifically, we should look for cases where the expression involves taking an even root of a negative number or any other operation that is undefined for some real numbers.

1. \( f(x) = x^3 \) is defined for all real numbers.
2. \( f(x) = (-x)^{\frac{1}{2}} \) involves taking the square root of a negative number, which is not defined for real numbers.
3. \( f(x) = (-x)^{\frac{1}{3}} \) is defined for all real numbers because cube roots of negative numbers are defined.
4. \( f(x) = 1 - |x|^{\frac{1}{2}} \) is defined for all real numbers because the square root of a non-negative number is defined.
5. \( f(x) = 1 - |x|^{\frac{1}{3}} \) is defined for all real numbers because cube roots of non-negative numbers are defined.

Thus, the function \( f(x) = (-x)^{\frac{1}{2}} \) is not a function for all values of \( x \).

We need to determine which of the given functions is not a function for all values of \( x \). A function is defined for all values of \( x \) if it produces a real number output for every real number input.

1. Function \( f(x) = x^3 \):

   • The cube of any real number is defined. Therefore, \( f(x) = x^3 \) is a function for all values of \( x \).

2. Function \( f(x) = (-x)^{\frac{1}{2}} \):

   • The square root of a negative number is not defined in the real number system. Therefore, \( f(x) = (-x)^{\frac{1}{2}} \) is not a function for all values of \( x \).

3. Function \( f(x) = (-x)^{\frac{1}{3}} \):

   • The cube root of a negative number is defined. Therefore, \( f(x) = (-x)^{\frac{1}{3}} \) is a function for all values of \( x \).

4. Function \( f(x) = 1 - |x|^{\frac{1}{2}} \):

   • The square root of an absolute value is always defined since \( |x| \) is non-negative. Therefore, \( f(x) = 1 - |x|^{\frac{1}{2}} \) is a function for all values of \( x \).

5. Function \( f(x) = 1 - |x|^{\frac{1}{3}} \):

   • The cube root of an absolute value is always defined. Therefore, \( f(x) = 1 - |x|^{\frac{1}{3}} \) is a function for all values of \( x \).

Final Answer