Final answer:
The end behavior of the function f(x) = 3x^3 + 2x^4 + x^2 + 1 is that it rises towards positive infinity as x approaches either positive or negative infinity, dictated by the positive leading coefficient and the even highest power.
Step-by-step explanation:
To determine the end behavior of a polynomial function like f(x) = 3x3 + 2x4 + x2 + 1, we look at the term with the highest degree because it will have the most significant impact on the function's behavior as x approaches positive or negative infinity. In this case, the term with the highest degree is 2x4. Since the leading coefficient (2) is positive and the highest power (4) is even, as x approaches positive or negative infinity, the function will rise towards positive infinity.
In conclusion, the end behavior of the given polynomial function is:
- As x approaches positive infinity, f(x) approaches positive infinity.
- As x approaches negative infinity, f(x) also approaches positive infinity.