To determine which function is even, we can use the definition of even functions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).
Let's analyze each function one by one:
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Function: \( f(x) = 6x^4 - 4x^2 + 8x - 2 \)
Check \( f(-x) \): \[ f(-x) = 6(-x)^4 - 4(-x)^2 + 8(-x) - 2 = 6x^4 - 4x^2 - 8x - 2 \] Since \( f(-x) \neq f(x) \) (specifically, the \( 8x \) term changes sign), this function is not even.
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Function: \( f(x) = 5x^6 - 3x^4 + 3x^2 - x \)
Check \( f(-x) \): \[ f(-x) = 5(-x)^6 - 3(-x)^4 + 3(-x)^2 - (-x) = 5x^6 - 3x^4 + 3x^2 + x \] Since \( f(-x) \neq f(x) \), this function is not even.
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Function: \( f(x) = \frac{-x^2}{x^4 - 25} \)
Check \( f(-x) \): \[ f(-x) = \frac{-(-x)^2}{(-x)^4 - 25} = \frac{-x^2}{x^4 - 25} \] Since \( f(-x) = f(x) \), this function is even.
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Function: \( f(x) = \frac{-6}{3x^3 - 25x} \)
Check \( f(-x) \): \[ f(-x) = \frac{-6}{3(-x)^3 - 25(-x)} = \frac{-6}{-3x^3 + 25x} = \frac{6}{3x^3 - 25x} \] Since \( f(-x) \neq f(x) \), this function is not even.
Based on this analysis, the only even function in the list is:
\( f(x) = \frac{-x^2}{x^4 - 25} \).