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The problem asks which of the following expressions is **not a function for all values of \( x \)**:

1. \( f(x) = x^2 \)
2. \( f(x) = \frac{1}{x} \)
3. \( f(x) = \frac{1}{x^2} \)
4. \( f(x) = \frac{1}{x^{1/2}} \)
5. \( f(x) = \frac{1}{x^3} \)

### Analysis:
- **Option 1**: \( f(x) = x^2 \) is a function for all real values of \( x \). It's a polynomial, which is always continuous and defined for all real numbers.
  
- **Option 2**: \( f(x) = \frac{1}{x} \) is undefined when \( x = 0 \), since division by zero is undefined. Therefore, this is **not** a function for all values of \( x \).

- **Option 3**: \( f(x) = \frac{1}{x^2} \) is undefined when \( x = 0 \) for the same reason as in Option 2. Therefore, this is **not** a function for all values of \( x \).

- **Option 4**: \( f(x) = \frac{1}{x^{1/2}} \) is undefined when \( x \leq 0 \) since the square root function only accepts non-negative inputs. Therefore, this is **not** a function for all real values of \( x \).

- **Option 5**: \( f(x) = \frac{1}{x^3} \) is undefined when \( x = 0 \) due to division by zero. So, this is **not** a function for all values of \( x \).

### Conclusion:
The correct answer is **Option 4**, \( f(x) = \frac{1}{x^{1/2}} \), because it is not defined for \( x \leq 0 \), whereas the other options only fail at \( x = 0 \).

Would you like further explanation on any part of this? Here are 5 related questions you might explore:

1. What is the domain of the square root function?
2. How do you determine when a rational function is undefined?
3. Can functions be defined piecewise to include or exclude certain values of \( x \)?
4. What is the difference between a polynomial and a rational function?
5. How does the behavior of functions change with negative exponents?

**Tip**: Always check the domain of a function by identifying values of \( x \) where the expression becomes undefined.

Which of the following is not a function for all values of x?
1. f(x) = x^2
2. f(x) = 1/x
3. f(x) = 1/x^2
4. f(x) = 1/x^(1/2)
5. f(x) = 1/x^3

Explore which functions are not defined for all values of x, including rational and radical functions. Learn about domain restrictions and undefined values, focusing on key concepts like polynomials and rational functions. Suitable for high school students in Grades 10-12.

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