How do you know if a function is even or odd algebraically?

Functions are the backbone of topics like calculus and trigonometry. Functions are of many types like even and odd.

Answer: Algebraially a function is an even function if  f(-x) = f(x), and it is a odd function if f(-x) = -f(x).

Let's understand the solution in detail.

Explanation:

Let us consider a function f(x).

Then, if you substitute x with -x in the function and the value of the function becomes negative, then the function is called an odd function.

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

For example, if f(x) = sin x and f(-x) = - sin x. Hence it follows the relation above. Hence, sin x is an odd function.

If we substitute x with -x in the function and the value of function does not change then the function is known even function.

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

For example, if f(x) = cos x and f(-x) = cos x. Hence it follows the relation above. Hence, cos x is an even function.

If the value of function does not fulfill the above conditions after substitution with -x, then the function is neither even function nor odd function.

Hence a function is an even function if f(-x) = f(x), and it is a odd function if f(-x) = -f(x).