Which of the following functions is odd?

Which of the following functions is odd?

• A. f(x) = x^3
• B. f(x) = x^2
• C. f(x) = |x|
• D. f(x) = e^x

Answer: (A)

An odd function has symmetry about the origin, meaning f(-x) = -f(x). For f(x) = x^3, f(-x) = (-x)^3 = -x^3 = -f(x), so this one is odd. For f(x) = x^2, f(-x) = (-x)^2 = x^2 = f(x), which is even, not odd. For f(x) = |x|, f(-x) = |-x| = |x| = f(x), also even. For f(x) = e^x, f(-x) = e^{-x}, which is not equal to -e^x, so it’s not odd (it’s neither even nor odd). Therefore, the odd function is f(x) = x^3.

An odd function has symmetry about the origin, meaning f(-x) = -f(x).

For f(x) = x^3, f(-x) = (-x)^3 = -x^3 = -f(x), so this one is odd.

For f(x) = x^2, f(-x) = (-x)^2 = x^2 = f(x), which is even, not odd.

For f(x) = |x|, f(-x) = |-x| = |x| = f(x), also even.

For f(x) = e^x, f(-x) = e^{-x}, which is not equal to -e^x, so it’s not odd (it’s neither even nor odd).

Therefore, the odd function is f(x) = x^3.