$$ \mathrm{f}(\mathrm{x})=\frac{1}{\mathrm{x}+7} $$ Choose the correct answer below. A. The y-intercept is (0,7). B. The horizontal asymptote is $$ \mathrm{y}=1 $$. C. The graph has an oblique asymptote. D. The $$ x $$-axis is its horizontal asymptote F. The $$ y $$-axis is its vertical asymptote. The vertical asymptote is $$ \mathrm{x}=1 $$. G. There is a hole in its graph at $$ \mathrm{x}=-7 $$. H. The x -intercept is $$ (-6,0) $$.

Question

UpStudy AI · Accepted Answer

Let's analyze each statement for the function $$ f(x) = \frac{1}{x + 7} $$:

1. **Y-Intercept (Option A)**:
   - To find the y-intercept, set $$ x = 0 $$:
     $$
     f(0) = \frac{1}{0 + 7} = \frac{1}{7}
     $$
     - **Statement A** claims the y-intercept is $$ (0, 7) $$, which is **incorrect**.

2. **Horizontal Asymptote (Options B & D)**:
   - As $$ x $$ approaches infinity, $$ f(x) $$ approaches 0:
     $$
     \lim_{x \to \pm\infty} \frac{1}{x + 7} = 0
     $$
     - **Statement B** claims the horizontal asymptote is $$ y = 1 $$, which is **incorrect**.
     - **Statement D** correctly identifies the horizontal asymptote as the x-axis ($$ y = 0 $$), which is **correct**.

3. **Oblique Asymptote (Option C)**:
   - Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. Here, both degrees are:
     - Numerator: 0 (constant)
     - Denominator: 1
     - Since the degree of the numerator is less, there is **no oblique asymptote**.
     - **Statement C** is **incorrect**.

4. **Vertical Asymptote and Holes (Options F & G)**:
   - **Vertical Asymptote**: Set the denominator to zero:
     $$
     x + 7 = 0 \implies x = -7
     $$
     - **Statement F** incorrectly states that the vertical asymptote is at $$ x = 1 $$ and that the y-axis is the vertical asymptote. Both parts are **incorrect**.
   - **Holes**: A hole occurs if there is a common factor in the numerator and denominator, which is not the case here.
     - **Statement G** is **incorrect**.

5. **X-Intercept (Option H)**:
   - To find the x-intercept, set $$ f(x) = 0 $$:
     $$
     \frac{1}{x + 7} = 0
     $$
     - This equation has **no solution**, meaning there is **no x-intercept**.
     - **Statement H** is **incorrect**.

**Conclusion:** Only **Option D** is correct.

**Answer:**  
D. The x -axis is its horizontal asymptote.
The correct answer is D: The x-axis is its horizontal asymptote.

Real Tutor Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions