Which equation defines an odd function?

Which equation defines an odd function?

• A. f(-x) = -f(x)
• B. f(x) = f(-x)
• C. f(-x) = f(x) - 2
• D. f(x) = 0

Answer: (A)

An odd function has symmetry about the origin: when you replace x with -x, the output becomes the negative of the original value. The equation f(-x) = -f(x) captures exactly that behavior, so it defines an odd function for all x in the domain. The other ideas describe different ideas of symmetry or are special cases. f(x) = f(-x) describes even symmetry, where flipping x doesn’t change the value. The relation f(-x) = f(x) - 2 doesn’t express a consistent symmetry about the origin. And f(x) = 0 is a specific function that happens to satisfy the odd condition as a special case, but it doesn’t serve as the general defining relation for odd functions—the defining relation is the first one.

An odd function has symmetry about the origin: when you replace x with -x, the output becomes the negative of the original value. The equation f(-x) = -f(x) captures exactly that behavior, so it defines an odd function for all x in the domain.

The other ideas describe different ideas of symmetry or are special cases. f(x) = f(-x) describes even symmetry, where flipping x doesn’t change the value. The relation f(-x) = f(x) - 2 doesn’t express a consistent symmetry about the origin. And f(x) = 0 is a specific function that happens to satisfy the odd condition as a special case, but it doesn’t serve as the general defining relation for odd functions—the defining relation is the first one.