An even function has which property?

• A. f(-x) = -f(x).
• B. f(-x) = f(x).
• C. Graph has a cusp at x=0.
• D. Domain is only nonnegative.

Answer: (B)

Evenness means the graph looks the same on both sides of the y-axis. That symmetry translates to f(-x) = f(x) for every x in the domain. So, replacing x with -x gives the same output. For example, f(x) = x^2 satisfies f(-x) = (-x)^2 = x^2, so it’s even. The other idea, f(-x) = -f(x), describes an odd function, where values flip sign when x becomes -x. A cusp at x = 0 is a matter of smoothness, not symmetry, and having a domain only on nonnegative values would prevent evaluating at -x, so it can’t define an even function. Therefore, the defining property is f(-x) = f(x).