Let's break down the function step by step:
$f(x) = \frac{5}{-(x + 1)}$
This is a rational function where the numerator is a constant and the denominator is linear. To analyze the features of its graph, we need to address each feature one by one:

1. Horizontal Asymptotes

For rational functions, horizontal asymptotes are determined by the behavior of the function as $x \to \infty$ or $x \to -\infty$. The horizontal asymptote depends on the degrees of the numerator and denominator:

• The numerator is a constant ($5$).
• The denominator is linear ($-(x + 1)$).

Since the degree of the denominator is higher than the numerator, the horizontal asymptote is at $y = 0$. This means as $x \to \infty$ or $x \to -\infty$, $f(x) \to 0$.

Conclusion: The function has 1 horizontal asymptote at $y = 0$.

2. Holes

Holes in the graph of a function occur when there is a factor that cancels out between the numerator and denominator. However, there are no factors that can cancel out here, as the numerator is simply $5$ and the denominator is $-(x + 1)$.

Conclusion: There are no holes in the graph.

3. Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero (and the numerator is non-zero).

The denominator is $-(x + 1)$. Set the denominator equal to zero to find potential vertical asymptotes: $-(x + 1) = 0$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$ Thus, the function has a vertical asymptote at $x = -1$.

Conclusion: The function has 1 vertical asymptote at $x = -1$.

4. x-intercepts

The x-intercepts of a function are the values of $x$ where $f(x) = 0$. For a rational function, this occurs when the numerator is zero (provided the denominator is not zero at the same point).

The numerator is $5$, which is never zero.

Conclusion: The function has no x-intercepts.

5. y-intercepts

The y-intercept is found by evaluating the function at $x = 0$.

Substitute $x = 0$ into the function: $f(0) = \frac{5}{-(0 + 1)} = \frac{5}{-1} = -5$ Thus, the y-intercept is $y = -5$.

Conclusion: The function has 1 y-intercept at $(0, -5)$.

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Summary of Features:

• Number of Horizontal Asymptotes: 1 (at $y = 0$)
• Number of Holes: None
• Number of Vertical Asymptotes: 1 (at $x = -1$)
• Number of x-intercepts: None
• Number of y-intercepts: 1 (at $(0, -5)$)

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Would you like more details on any part of this analysis or any specific aspect clarified? Also, here are some related questions:

1. How does the position of the vertical asymptote affect the graph’s behavior?
2. What happens to the function’s graph as $x$ approaches the horizontal asymptote?
3. How can we classify rational functions based on their degrees for asymptotic behavior?
4. What are the general steps to finding the intercepts of a rational function?
5. How do we graph rational functions with both horizontal and vertical asymptotes?

Tip: Always check if any factors in the numerator and denominator can cancel out to identify potential holes in the graph.