Final answer:
The function g(x) = x³ - 9x is an odd function because it satisfies the condition g(-x) = -g(x) and is anti-symmetric about the origin. It is neither even nor neither because the condition g(-x) = g(x) is not met.
Step-by-step explanation:
To determine if the function g(x) = x³ - 9x is even, odd, or neither, we apply the definitions of even and odd functions. An even function satisfies the condition f(-x) = f(x), meaning it is symmetric about the y-axis. Conversely, an odd function satisfies the condition f(-x) = -f(x), indicating that it has point symmetry about the origin.
To test if g(x) is even, we replace x with -x: g(-x) = (-x)³ - 9(-x) = -x³ + 9x.
This result does not satisfy g(-x) = g(x); therefore, g(x) is not an even function.
To test if g(x) is odd, we again use g(-x) and check if it equals -g(x): -g(x) = - (x³ - 9x) = -x³ + 9x, which is equal to g(-x).
Since g(-x) = -g(x), the function g(x) is indeed an odd function. Therefore, it is anti-symmetric about the origin, and the integral of g(x) over all space (from -infinity to +infinity) will be zero, satisfying another characteristic of odd functions.