How to Determine if a Function is Odd, Even, or Neither

To determine whether a function is odd, even, or neither, you need to analyze its properties concerning symmetry.

Step 1: Understand the Definitions

• Even Function: A function f(x) is even if for every x in the domain, f(-x) = f(x). This means the function is symmetric about the y-axis.
• Odd Function: A function f(x) is odd if for every x in the domain, f(-x) = -f(x). This means the function has rotational symmetry about the origin.
• Neither: If a function fails to meet the criteria for being even or odd, it is categorized as neither.

Step 2: Test the Function

To check if a function is even or odd, follow these steps:

1. Substitute -x into the function and simplify.
2. If you find that f(-x) = f(x), then the function is even.
3. If you find that f(-x) = -f(x), then the function is odd.
4. If neither condition holds, then the function is neither odd nor even.

Example

Consider the function f(x) = x^3 – x.

1. Calculate f(-x):
f(-x) = (-x)^3 – (-x) = -x^3 + x.

2. Compare with f(x):
f(-x) = -x^3 + x which is not equal to f(x) or -f(x) (where -f(x) = -x^3 + x).

3. Since neither condition for odd or even is met, f(x) is neither odd nor even.

Conclusion

By substituting -x into the function and comparing the results, you can effectively classify any function as odd, even, or neither based on its symmetry. This approach works for polynomial functions, trigonometric functions, and many others.