Final answer:
Among the given functions, f(x) = -x^2 / (x^4 - 25) is the even function because it satisfies the condition f(x) = f(-x), making it symmetric about the y-axis.
Step-by-step explanation:
The question asks us to identify which of the given functions is even. An even function is one that satisfies the condition f(x) = f(-x), which essentially means that the function is symmetric about the y-axis. The function is even if, when we substitute -x for x, the function remains unchanged. Let's analyze the given options one by one:
f(x) = 6x^4 - 4x^2 + 8x - 2 is not an even function because the term 8x changes sign when x is replaced by -x, thus f(-x) ≠ f(x).f(x) = 5x^6 - 3x^4 + 3x^2 - x is also not even for the same reason; the -x term changes sign when x is replaced by -x.f(x) = -x^2 / (x^4 - 25) simplifies to -1 / (x^2 + 25), which is an even function because when we substitute -x for x, the resultant expression does not change, so f(-x) = f(x).f(x) = -6 / (3x^3 - 25x) is not even because replacing x with -x will change the sign of the terms in the denominator, hence f(-x) ≠ f(x).
The correct answer is the third function, f(x) = -x^2 / (x^4 - 25).