Which statement describes a function that is neither odd nor even?

Which statement describes a function that is neither odd nor even?

• A. Mixture of odd and even exponents
• B. Odd function shifted in any direction
• C. Even function shifted left or right
• D. Only even exponents

Answer: (A)

Think about symmetry: a function is odd if f(-x) = -f(x) and even if f(-x) = f(x). If a function has both odd and even powers in its expression, the two kinds of symmetry don’t line up, so the whole function doesn’t satisfy either condition. That’s why a mix of odd and even exponents describes a function that is neither. For example, if you have f(x) = x^3 + x^2, then f(-x) = -x^3 + x^2, which is not equal to f(x) and not equal to -f(x). The combination of both types of terms breaks both symmetries. The other ideas talk about shifting an odd or an even function. Shifting can destroy the original symmetry, so stating that an odd function is shifted in any direction or that an even function is shifted left or right does not capture a function that is necessarily neither in the same clear way as having both odd and even terms does.

Think about symmetry: a function is odd if f(-x) = -f(x) and even if f(-x) = f(x). If a function has both odd and even powers in its expression, the two kinds of symmetry don’t line up, so the whole function doesn’t satisfy either condition. That’s why a mix of odd and even exponents describes a function that is neither.

For example, if you have f(x) = x^3 + x^2, then f(-x) = -x^3 + x^2, which is not equal to f(x) and not equal to -f(x). The combination of both types of terms breaks both symmetries.

The other ideas talk about shifting an odd or an even function. Shifting can destroy the original symmetry, so stating that an odd function is shifted in any direction or that an even function is shifted left or right does not capture a function that is necessarily neither in the same clear way as having both odd and even terms does.