Which of the following describes the end behavior of the function f(x) = -2x^3 + 5x - 1 as x approaches positive infinity?

Question

Accepted Answer

f(x) approaches negative infinity. The end behavior of a polynomial is determined by its leading term  -  the term with the highest degree. For f(x) = -2x^3 + 5x - 1, the leading term is -2x^3. As x → +∞, x^3 → +∞, so -2x^3 → -∞. Therefore, f(x) → -∞ as x → +∞. The lower-degree terms (5x and -1) become negligible compared to -2x^3 for very large values of x.

The end behavior of a polynomial depends on two factors: the degree (odd or even) and the sign of the leading coefficient (positive or negative). For an odd-degree polynomial with a negative leading coefficient (like -2x^3): as x → +∞, f(x) → -∞, and as x → -∞, f(x) → +∞. The ends go in opposite directions, and they are reversed from the standard positive cubic (which goes from bottom-left to top-right).

"f(x) approaches positive infinity" would be correct if the leading coefficient were positive (like +2x^3). "f(x) approaches 0" (approaching 0) describes the end behavior of certain rational functions, not polynomials  -  polynomials always approach ±∞ as x → ±∞ (except for constant polynomials). "f(x) approaches -1" (approaching -1) confuses the constant term with the end behavior  -  the y-intercept is f(0) = -1, but this tells us nothing about the function's behavior as x becomes very large. Understanding polynomial end behavior is important for graphing and analyzing functions on the CLEP exam.

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