To determine the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 3x^2 + 5x - 1 \)[/tex], we need to analyze the term that has the highest power, which dominates the behavior of the polynomial as [tex]\( x \to +\infty \)[/tex] and [tex]\( x \to -\infty \)[/tex].
1. Identify the highest degree term:
The polynomial function provided is [tex]\( f(x) = 2x^3 - 3x^2 + 5x - 1 \)[/tex]. The highest degree term here is [tex]\( 2x^3 \)[/tex].
2. Analyze the behavior as [tex]\( x \to +\infty \)[/tex]:
When [tex]\( x \to +\infty \)[/tex], the [tex]\( x^3 \)[/tex] term will dominate the polynomial because it grows faster than the other terms. Since the coefficient of [tex]\( x^3 \)[/tex] is positive (i.e., [tex]\( 2 \)[/tex]), as [tex]\( x \to +\infty \)[/tex], the [tex]\( 2x^3 \)[/tex] term will also grow to [tex]\( +\infty \)[/tex]. Therefore, [tex]\( f(x) \to +\infty \)[/tex] as [tex]\( x \to +\infty \)[/tex].
3. Analyze the behavior as [tex]\( x \to -\infty \)[/tex]:
Similarly, when [tex]\( x \to -\infty \)[/tex], the [tex]\( x^3 \)[/tex] term still dominates the behavior of the polynomial. However, the key difference is that the term [tex]\( x^3 \)[/tex] will now take very large negative values because raising a negative number to an odd power results in a negative number. Since the coefficient of [tex]\( x^3 \)[/tex] is positive, as [tex]\( x \to -\infty \)[/tex], the [tex]\( 2x^3 \)[/tex] term will grow to [tex]\( -\infty \)[/tex]. Therefore, [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Combining these observations, the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 3x^2 + 5x - 1 \)[/tex] is as follows:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
In conclusion, the correct identification of the end behavior for the polynomial function is:
[tex]\[
\begin{array}{r}
f(x) = 2x^3 - 3x^2 + 5x - 1 \\
f(x) \to -\infty \text{ as } x \to -\infty \\
f(x) \to +\infty \text{ as } x \to +\infty
\end{array}
\][/tex]
