Final answer:
Option A. To determine whether the function is even, odd, or neither, we check its symmetry. In this case, the function is even. Therefore, the answer is A. Even.
Step-by-step explanation:
To determine algebraically whether the given function f(x) = -9x³ is even, odd, or neither, we need to review the definitions of even and odd functions. An even function is characterized by the property f(x) = f(-x), meaning that the function is symmetric about the y-axis. On the other hand, an odd function has the property f(-x) = -f(x), which implies that the function is symmetric about the origin.
Applying this to the function f(x) = -9x³, let's replace x with -x and observe the outcome: f(-x) = -9(-x)³ = -9(-1)³x³ = 9x³ Since f(-x) = -f(x), the given function is an odd function.
To determine whether the function is even, odd, or neither, we need to analyze its symmetry. A function is considered even if it satisfies the condition f(x) = f(-x) for all x in its domain, meaning that it is symmetric with respect to the y-axis. On the other hand, a function is odd if it satisfies the condition f(x) = -f(-x) for all x in its domain, meaning that it is symmetric with respect to the origin.
In this case, the function is f(x) = -9x³. To check if it is even, we substitute -x for x:
f(-x) = -9(-x)³ = -9(-x²)(-x) = -9x²(-x) = -9x³ = f(x)
Since f(x) = f(-x), the function is even. Therefore, the answer is A. Even.