What describes the end behavior of the polynomial function f(x) = x^3?

What describes the end behavior of the polynomial function f(x) = x^3?

• A. f(x) approaches zero as x approaches infinity
• B. f(x) approaches infinity as x approaches both infinity and negative infinity
• C. f(x) approaches negative infinity as x approaches positive infinity
• D. f(x) approaches infinity as x approaches infinity and negative infinity as x approaches negative infinity

Answer: (D)

The end behavior of the polynomial function f(x) = x^3 can be analyzed by looking at its leading term, which in this case is x^3. As x approaches positive infinity (x → ∞), the value of f(x) also approaches positive infinity. This happens because larger positive numbers raised to the third power yield larger and larger positive results. Conversely, as x approaches negative infinity (x → -∞), the value of f(x) approaches negative infinity. This is due to the fact that raising larger negative numbers to an odd power, such as three, results in increasingly large negative outcomes. For example, (-10)^3 equals -1000, which is quite negative, and similarly for other large negative inputs. Thus, the correct answer describes that f(x) approaches infinity when x approaches positive infinity, and approaches negative infinity when x approaches negative infinity. This behavior is typical of odd-degree polynomial functions, where the ends of the graph will rise on one side and fall on the other.

The end behavior of the polynomial function f(x) = x^3 can be analyzed by looking at its leading term, which in this case is x^3. As x approaches positive infinity (x → ∞), the value of f(x) also approaches positive infinity. This happens because larger positive numbers raised to the third power yield larger and larger positive results.

Conversely, as x approaches negative infinity (x → -∞), the value of f(x) approaches negative infinity. This is due to the fact that raising larger negative numbers to an odd power, such as three, results in increasingly large negative outcomes. For example, (-10)^3 equals -1000, which is quite negative, and similarly for other large negative inputs.

Thus, the correct answer describes that f(x) approaches infinity when x approaches positive infinity, and approaches negative infinity when x approaches negative infinity. This behavior is typical of odd-degree polynomial functions, where the ends of the graph will rise on one side and fall on the other.