An odd function has which property?

An odd function has which property?

• A. f(-x) = f(x).
• B. f(-x) = -f(x).
• C. Graph is symmetric about the y-axis.
• D. Function is defined only for x>0.

Answer: (B)

An odd function is one where plugging in the negative input flips the sign of the output: f(-x) = -f(x) for all x in the domain. This property makes the graph symmetric with respect to the origin (rotate 180 degrees around the origin and it looks the same). The statement that f(-x) equals the negative of f(x) exactly captures this defining behavior, so it’s the best choice. The other ideas describe different ideas. If f(-x) = f(x), that’s an even function with symmetry about the y-axis, not about the origin. Saying the graph is symmetric about the y-axis points to that evenness. And being defined only for x > 0 would prevent the function from exhibiting the necessary behavior across negative x, which odd functions require.

An odd function is one where plugging in the negative input flips the sign of the output: f(-x) = -f(x) for all x in the domain. This property makes the graph symmetric with respect to the origin (rotate 180 degrees around the origin and it looks the same). The statement that f(-x) equals the negative of f(x) exactly captures this defining behavior, so it’s the best choice.

The other ideas describe different ideas. If f(-x) = f(x), that’s an even function with symmetry about the y-axis, not about the origin. Saying the graph is symmetric about the y-axis points to that evenness. And being defined only for x > 0 would prevent the function from exhibiting the necessary behavior across negative x, which odd functions require.