The end behavior of a polynomial function like f(x) = 3x^4 + x^3 – 2x^2 + 4x – 5 is primarily determined by its leading term, which is the term with the highest degree. In this case, the leading term is 3x^4.
As x approaches positive infinity (x → +∞):
- Since the leading term 3x^4 is positive and the degree (4) is even, the function f(x) will also approach positive infinity. Hence, we can say that:
f(x) → +∞ as x → +∞
As x approaches negative infinity (x → -∞):
- Similarly, because the leading term 3x^4 is positive and the degree is even, the function f(x) will also approach positive infinity in this direction. Therefore:
f(x) → +∞ as x → -∞
In summary, the end behavior of the polynomial function f(x) = 3x^4 + x^3 – 2x^2 + 4x – 5 can be described as:
- As x approaches +∞, f(x) approaches +∞.
- As x approaches -∞, f(x) approaches +∞.
This indicates that both ends of the graph of the function will rise toward positive infinity, showing a characteristic of even-degree polynomial functions with a positive leading coefficient.