Final answer:
A function is said to be even if it satisfies the property f(x) = f(-x) for all x in the domain of the function. A function is said to be odd if it satisfies the property f(x) = -f(-x) for all x in the domain of the function. A function is neither even nor odd if it does not satisfy either of these properties.
Step-by-step explanation:
A function is said to be even if it satisfies the property f(x) = f(-x) for all x in the domain of the function. On the other hand, a function is said to be odd if it satisfies the property f(x) = -f(-x) for all x in the domain of the function. Lastly, a function is said to be neither even nor odd if it does not satisfy either of these properties. To determine whether a function is even, odd, or neither, we need to check whether these properties hold true.
Let's consider an example:
Let f(x) = x^3 - x^2. To determine if this function is even, odd, or neither, we need to check the properties f(x) = f(-x) and f(x) = -f(-x). Let's evaluate these properties:
f(x) = x^3 - x^2
f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2
Since f(x) is not equal to f(-x), the function is not even. Now let's check the property f(x) = -f(-x):
f(x) = x^3 - x^2
-f(-x) = -(-x^3 - x^2) = x^3 + x^2
Again, f(x) is not equal to -f(-x), so the function is not odd either. Therefore, the function f(x) = x^3 - x^2 is neither even nor odd.