How to determine if a function is odd, even or neither?

We will use the concept of odd and even functions to find the nature of the function.

Answer: If f(x) = - f(-x), then f is an odd function. If f(x) = f(-x), then f is an even function. If neither of these conditions hold, then f is neither even nor odd function.

Let use the definition of even and odd function to answer this question.

Explanation:

Let us consider a function f(x).

Then, if we substitute x with -x in the function and the value of function becomes negative, then the function is known as an odd function. For example, f(x) = 2x is odd function.

Hence, for odd function, f(x) = - f(-x)

If we substitute x with -x in the function and the value of function does not change, then the function is known even function. For example, f(x) = x² is even function.

Hence, for even function, f(x) = f(-x) 

If we substitute x with -x in the function and it neither satisfies f(x) = - f(-x) nor f(x) =  f(-x), then the function is neither even function nor odd function. For example, f(x) = x² + x is neither even nor odd function.

Therefore, If f(x) = - f(-x), then f is an odd function and if f(x) = f(-x), then f is an even function.