The end behavior of a polynomial function gives us a glimpse into how the function behaves as the input values (x) approach positive or negative infinity. For the function f(x) = x³ – 2x² + 4x – 5, the dominant term is x³, because it has the highest degree.
As x approaches positive infinity (x → +∞), the x³ term will dominate the other terms, which means:
- All other terms become negligible when compared to x³.
- Thus, we can approximate that f(x) approaches +∞.
Therefore, we can say:
As x → +∞, f(x) → +∞
On the flip side, as x approaches negative infinity (x → -∞), the behavior changes due to the odd degree of the dominant term:
- Here, x³ will still dominate, but as we input negative values, the result will also be negative.
- Thus, we find that f(x approaches -∞.
In summary:
As x → -∞, f(x) → -∞
Putting it all together, the end behavior of the function f(x) = x³ – 2x² + 4x – 5 can be succinctly stated as:
- f(x) → +∞ as x → +∞
- f(x) → -∞ as x → -∞
This end behavior is crucial for sketching the graph of the polynomial function, aiding in understanding how the function will rise or fall at its extremes.