Explanation:
End behavior describes the behavior of a function as the input approaches infinity or negative infinity.
For the function f(x) = -log_3(x^3), let's first consider the behavior as x approaches infinity, and then as x approaches 0 (from the positive side, since log is undefined for non-positive inputs).
When x approaches infinity (x → ∞):
- The inner function x^3 also approaches infinity.
- The logarithm function increases without bound as its input approaches infinity. This means log_3(x^3) also increases without bound.
- However, because there is a negative sign in front of the logarithm, the function f(x) will actually decrease without bound. Hence, f(x) → -∞ as x → ∞.
Now, consider the behavior as x approaches 0 from the right (x → 0^+):
- The inner function x^3 approaches 0.
- The logarithm function approaches negative infinity as its input approaches 0 from the right. This means log_3(x^3) → -∞ as x → 0^+.
- Therefore, the function f(x), being the negative of log_3(x^3), will approach positive infinity. So, f(x) → ∞ as x → 0^+.
In conclusion, the end behaviors of the function f(x) = -log_3(x^3) are:
- As x → ∞, f(x) → -∞.
- As x → 0^+, f(x) → ∞.