If f(x)=f(−x) for all x, then f(x) could NOT be

Rationale

(x^3-1)^2

The function f(x) = (x^3 - 1)^2 does not satisfy the condition f(x) = f(-x) for all x, indicating it is not an even function. This is because the term x^3 introduces asymmetry, causing the function to yield different values for positive and negative inputs.

A (x²-2) YOUR ANSWER

This function is even since f(-x) = (-x)² - 2 = x² - 2, which equals f(x). The symmetry about the y-axis confirms that f(x) = f(-x) for all x, meeting the required condition.

B (x²+2) YOUR ANSWER

Similarly, this function is also even. We find that f(-x) = (-x)² + 2 = x² + 2, which is equal to f(x). Thus, it satisfies the property f(x) = f(-x) for all x.

C (x^4-x²) YOUR ANSWER

This function is even as well. The calculation shows f(-x) = (-x)^4 - (-x)² = x^4 - x², which equals f(x). Therefore, it adheres to the condition f(x) = f(-x) for all x.

D ((x^3-1)^2) CORRECT

This function is not even because f(-x) = ((-x)^3 - 1)² = (-x^3 - 1)², which expands differently than f(x). The cubic term x^3 introduces asymmetry, leading to f(x) ≠ f(-x) for all x.

E ((x^3-x)^2) YOUR ANSWER

This function is even as well. When calculating f(-x) = ((-x)^3 - (-x))² = (x^3 + x)², we find it equals f(x). Thus, it maintains the required symmetry and satisfies f(x) = f(-x) for all x.

Conclusion

The requirement f(x) = f(-x) indicates that f(x) must be an even function. While functions A, B, C, and E fulfill this condition and exhibit symmetry, function D fails to meet the criterion due to the asymmetrical influence of the cubic term. Hence, f(x) = (x^3 - 1)² could not satisfy the even function property.

If f(x)=f(−x) for all x, then f(x) could NOT be

(x^3-1)^2

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