To determine if a function is even or odd, you can follow these algebraic steps:
1. Identify the function: Start with your function, let’s call it f(x).
2. Test for evenness: To check if the function is even, evaluate f(-x). If f(-x) = f(x) for all x in the domain of the function, then the function is even. Even functions are symmetric with respect to the y-axis.
Example: Consider f(x) = x^2. We find f(-x) = (-x)^2 = x^2 = f(x), confirming that f(x) is even.
3. Test for oddness: To check if the function is odd, again evaluate f(-x). If f(-x) = -f(x) for all x in the domain of the function, then the function is odd. Odd functions exhibit symmetry with respect to the origin.
Example: Take f(x) = x^3. We calculate f(-x) = (-x)^3 = -x^3 = -f(x), which shows that f(x) is odd.
4. Conclusion: If neither condition is satisfied, the function is neither even nor odd. It is also important to note that a function can be both even and odd only if it is the zero function (f(x) = 0).