Final answer:
The end behavior of the function f(x) = -8x^4 - 2x^3 + x is that the graph falls to negative infinity as x approaches both positive and negative infinity.
Step-by-step explanation:
The end behavior of the graph of the function f(x) = -8x⁴ - 2x³ + x is described by the highest degree term, which is -8x⁴. Since the leading coefficient is negative and the highest degree is even, the graph falls to negative infinity on both ends.
This means as x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) also approaches negative infinity. Therefore, the correct answers are: A. As x → [infinity], f(x) → -[infinity], and C. As x → -[infinity], f(x) → -[infinity].
The end behavior of a function determines what happens to the function as the input approaches positive or negative infinity. For the given function f(x) = -8x⁴ - 2x³ + x, the leading term is -8x⁴, which has an even degree and a negative coefficient.
This means that as x approaches positive or negative infinity, the function f(x) will also approach negative infinity, making option A the correct answer.