Question
Find the horizontal and vortical asymptotes of \( f(x) \). \( f(x)=\frac{7 x}{x+4} \) Find the horizontal asymptotes. Select the correct choice and, if necessary, fill in the answer box(es) to completo your choice. A. The function has one horizontal asymptote, \( y=7 \). (Type an equation.) B. The function has two horizontal asymptotes. The top asymplole is \( \square \) and the bottom asymplote is (Type equations.) The function has no horizontal asymptotes. Find the vertical asymptotes. Select the correct choice and, if necessary, fill in the answer box(es) to complele your choice A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is \( \square \) and the rightmost asymptote is (Type equations.) C. The function has no vertical asymptotes.
Ask by Gray Parry. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
- **Horizontal Asymptote:** \( y = 7 \)
- **Vertical Asymptote:** \( x = -4 \)
Solution
To analyze the asymptotes of the function \( f(x) = \frac{7x}{x+4} \), we'll examine both horizontal and vertical asymptotes separately.
### Horizontal Asymptotes
**Definition:** Horizontal asymptotes describe the behavior of a function as \( x \) approaches \( \pm\infty \).
For the function \( f(x) = \frac{7x}{x+4} \):
1. **Compare Degrees:** Both the numerator and the denominator are polynomials of degree 1.
2. **Leading Coefficients:** The leading coefficient of the numerator is 7, and the leading coefficient of the denominator is 1.
3. **Horizontal Asymptote:** When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
\[
y = \frac{7}{1} = 7
\]
**Conclusion:** There is **one horizontal asymptote** at \( y = 7 \).
**Choice:**
- **A.** The function has one horizontal asymptote, \( y = 7 \).
### Vertical Asymptotes
**Definition:** Vertical asymptotes occur where the denominator of a rational function is zero (and the numerator isn't zero at those points).
For the function \( f(x) = \frac{7x}{x+4} \):
1. **Set Denominator to Zero:**
\[
x + 4 = 0 \implies x = -4
\]
2. **Check the Numerator at \( x = -4 \):**
\[
7(-4) = -28 \neq 0
\]
Since the numerator is not zero at \( x = -4 \), there is a vertical asymptote at this point.
**Conclusion:** There is **one vertical asymptote** at \( x = -4 \).
**Choice:**
- **A.** The function has one vertical asymptote, \( x = -4 \).
### Summary
- **Horizontal Asymptote:** \( y = 7 \)
- **Vertical Asymptote:** \( x = -4 \)
**Final Answer:**
- **Horizontal asymptote:** A. The function has one horizontal asymptote, \( y = 7 \).
- **Vertical asymptote:** A. The function has one vertical asymptote, \( x = -4 \).
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Simplify this solution Extra Insights
To find the horizontal asymptote of \( f(x) = \frac{7x}{x+4} \), we analyze the degrees of the numerator and denominator. Both the numerator and denominator are of degree 1. Hence, the horizontal asymptote can be found by dividing the leading coefficients: \( \frac{7}{1} = 7 \). So, the function has one horizontal asymptote at \( y = 7 \). As for the vertical asymptote, we look for values of \( x \) that make the denominator equal to zero. Setting \( x + 4 = 0 \) gives us \( x = -4 \). Therefore, the function has one vertical asymptote at \( x = -4 \). In summary, the answers are: Horizontal asymptote: A. \( y = 7 \) Vertical asymptote: A. \( x = -4 \)
