To determine whether the function g(x) = (x²sin(x))/(x²-9) is even, odd, or neither, we can analyze its properties with respect to symmetry:
1. Even Function: A function is even if it satisfies the condition f(x) = f(-x) for all x in its domain.
2. Odd Function: A function is odd if it satisfies the condition f(x) = -f(-x) for all x in its domain.
Let's analyze g(x):
g(-x) = [(-x)²sin(-x)]/((-x)²-9)
= [x²(-sin(x))]/(x²-9)
Now, let's compare g(x) and g(-x):
g(x) = (x²sin(x))/(x²-9)
g(-x) = [x²(-sin(x))]/(x²-9)
As you can see, g(x) and g(-x) are not equal, and they are also not equal to the negation of each other. Therefore, g(x) does not satisfy the conditions for being an even or odd function. It is neither even nor odd.