Answer :
To determine the end behavior of the polynomial function [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex], we need to analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches very large positive and negative values.
### Step 1: Analyze the Leading Term
For polynomials, the highest degree term (leading term) dictates the end behavior. In this case, the leading term is [tex]\( x^5 \)[/tex]:
[tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex]
### Step 2: As [tex]\( x \to +\infty \)[/tex]
- When [tex]\( x \)[/tex] is a very large positive number, [tex]\( x^5 \)[/tex] dominates and [tex]\( x^5 \)[/tex] is positive.
- Therefore, [tex]\( f(x) \)[/tex] will also be a very large positive number.
So, we have:
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]
### Step 3: As [tex]\( x \to -\infty \)[/tex]
- When [tex]\( x \)[/tex] is a very large negative number, [tex]\( x^5 \)[/tex] dominates and [tex]\( x^5 \)[/tex] is negative.
- Therefore, [tex]\( f(x) \)[/tex] will also be a very large negative number.
So, we have:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
### Conclusion:
The end behavior of the graph can be described as follows:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]
Thus, the correct answer is:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty ; f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]
### Step 1: Analyze the Leading Term
For polynomials, the highest degree term (leading term) dictates the end behavior. In this case, the leading term is [tex]\( x^5 \)[/tex]:
[tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex]
### Step 2: As [tex]\( x \to +\infty \)[/tex]
- When [tex]\( x \)[/tex] is a very large positive number, [tex]\( x^5 \)[/tex] dominates and [tex]\( x^5 \)[/tex] is positive.
- Therefore, [tex]\( f(x) \)[/tex] will also be a very large positive number.
So, we have:
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]
### Step 3: As [tex]\( x \to -\infty \)[/tex]
- When [tex]\( x \)[/tex] is a very large negative number, [tex]\( x^5 \)[/tex] dominates and [tex]\( x^5 \)[/tex] is negative.
- Therefore, [tex]\( f(x) \)[/tex] will also be a very large negative number.
So, we have:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
### Conclusion:
The end behavior of the graph can be described as follows:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]
Thus, the correct answer is:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty ; f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]