Determining whether a function is even, odd, or neither involves examining its symmetry and behavior under negation. Here’s how you can tell the difference:
Even Functions
A function f(x) is considered even if it satisfies the following condition:
- f(-x) = f(x)
This means that if you replace x with -x in the function, the output remains the same. Graphically, even functions are symmetric with respect to the y-axis. A classic example of an even function is f(x) = x2.
Odd Functions
A function f(x) is classified as odd if it meets the following criterion:
- f(-x) = -f(x)
In this case, replacing x with -x results in the negative of the original function’s output. Odd functions exhibit symmetry with respect to the origin. A well-known example is f(x) = x3.
Neither Even Nor Odd
If a function does not satisfy the criteria for being either even or odd, it is classified as neither. This means that neither the equality f(-x) = f(x) nor f(-x) = -f(x) holds true. For instance, the function f(x) = x + 1 does not fall into either category.
Steps to Determine Even, Odd, or Neither
- Substitute -x into the function to get f(-x).
- Compare f(-x) to f(x) and to -f(x).
- If f(-x) = f(x), then it’s even.
- If f(-x) = -f(x), then it’s odd.
- If neither condition is met, the function is neither even nor odd.
By following these steps and understanding the properties of even and odd functions, you can easily classify any function you encounter.