Which of the following statements is true for even functions?

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The statement indicating that f(-x) = f(x) is the defining characteristic of even functions. This relationship shows that for any input x, the function yields the same output whether x is positive or negative. This symmetry about the y-axis is what classifies a function as even; it means that the function's graph remains unchanged if flipped over the y-axis.

In contrast, the other statements describe different properties of functions. The first statement, which describes an odd function rather than an even function, asserts that the function produces outputs that are the negative of the inputs’ outputs when inputs are negated. The third option posits that f(x) is greater than zero for all x, which is not universally true for even functions; for instance, the function f(x) = x^2 is even, yet it only equals zero when x = 0. The fourth choice implies that f(x) is always increasing, which also does not apply to all even functions, as they may increase for some intervals while decreasing in others.

Thus, the correct identification of f(-x) = f(x) confirms that the function exhibits even symmetry, making it the accurate description of even functions.

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