Answer :
To determine the end behavior of the function \( f(x) = \frac{8x + 1}{2x - 9} \), we analyze the behavior of the function as \( x \) approaches both positive infinity (\( \infty \)) and negative infinity (\( -\infty \)).
### Step-by-Step Solution:
1. Understanding End Behavior:
The end behavior of a rational function \(\frac{P(x)}{Q(x)}\) is determined by the degrees of the numerator \(P(x)\) and the denominator \(Q(x)\). In this case:
- The degree of the numerator \(8x + 1\) is 1.
- The degree of the denominator \(2x - 9\) is also 1.
2. End Behavior as \( x \rightarrow \infty \):
We focus on the highest degree terms of the numerator and denominator, as they dominate the behavior of the function for large values of \(x\).
- The highest degree terms are \(8x\) in the numerator and \(2x\) in the denominator.
- So, we approximate the function as \( x \rightarrow \infty \):
[tex]\[ f(x) \approx \frac{8x}{2x} = 4 \][/tex]
Thus, as \( x \rightarrow \infty \), \( f(x) \rightarrow 4 \).
3. End Behavior as \( x \rightarrow -\infty \):
Similar to the positive infinity case, we focus on the highest degree terms for large negative values of \(x\).
- The highest degree terms are \(8x\) in the numerator and \(2x\) in the denominator.
- So, we approximate the function as \( x \rightarrow -\infty \):
[tex]\[ f(x) \approx \frac{8x}{2x} = 4 \][/tex]
Thus, as \( x \rightarrow -\infty \), \( f(x) \rightarrow 4 \).
### Conclusion:
From the analysis, we see that the end behavior of the function \( f(x) \) as \( x \) approaches both \( \infty \) and \( -\infty \) is the same. Both limits approach 4.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{As } x \rightarrow -\infty, f(x) \rightarrow 4 ; \text{ as } x \rightarrow \infty, f(x) \rightarrow 4.} \][/tex]
### Step-by-Step Solution:
1. Understanding End Behavior:
The end behavior of a rational function \(\frac{P(x)}{Q(x)}\) is determined by the degrees of the numerator \(P(x)\) and the denominator \(Q(x)\). In this case:
- The degree of the numerator \(8x + 1\) is 1.
- The degree of the denominator \(2x - 9\) is also 1.
2. End Behavior as \( x \rightarrow \infty \):
We focus on the highest degree terms of the numerator and denominator, as they dominate the behavior of the function for large values of \(x\).
- The highest degree terms are \(8x\) in the numerator and \(2x\) in the denominator.
- So, we approximate the function as \( x \rightarrow \infty \):
[tex]\[ f(x) \approx \frac{8x}{2x} = 4 \][/tex]
Thus, as \( x \rightarrow \infty \), \( f(x) \rightarrow 4 \).
3. End Behavior as \( x \rightarrow -\infty \):
Similar to the positive infinity case, we focus on the highest degree terms for large negative values of \(x\).
- The highest degree terms are \(8x\) in the numerator and \(2x\) in the denominator.
- So, we approximate the function as \( x \rightarrow -\infty \):
[tex]\[ f(x) \approx \frac{8x}{2x} = 4 \][/tex]
Thus, as \( x \rightarrow -\infty \), \( f(x) \rightarrow 4 \).
### Conclusion:
From the analysis, we see that the end behavior of the function \( f(x) \) as \( x \) approaches both \( \infty \) and \( -\infty \) is the same. Both limits approach 4.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{As } x \rightarrow -\infty, f(x) \rightarrow 4 ; \text{ as } x \rightarrow \infty, f(x) \rightarrow 4.} \][/tex]
