To analyze the function $y = \frac{5}{x + 4} - 5$, let's look at a few key features: domain, range, asymptotes, and the overall behavior of the function.

Step-by-Step Analysis

1. Domain:
   The function $y = \frac{5}{x + 4} - 5$ is undefined where the denominator equals zero, as division by zero is not possible.

   $x + 4 = 0 \Rightarrow x = -4$

   Therefore, the domain of the function is all real numbers except $x = -4$:

   $\text{Domain: } x \in \mathbb{R}, \, x \neq -4$

2. Range:
   The function has a vertical asymptote (where $x = -4$), and it shifts vertically by -5 due to the $-5$ term outside the fraction.

   We analyze the horizontal behavior by considering what happens as $x \to \infty$ and $x \to -\infty$.

   • As $x \to \infty$, $\frac{5}{x+4} \to 0$, so $y \approx -5$.
   • As $x \to -\infty$, $\frac{5}{x+4} \to 0$, so $y \approx -5$.

   Thus, the horizontal asymptote is $y = -5$, and $y$ can approach values arbitrarily close to $-5$ but never exactly reach it. Therefore, the range is:

   $\text{Range: } y \in \mathbb{R}, \, y \neq -5$

3. Asymptotes:

   • Vertical Asymptote: $x = -4$
   • Horizontal Asymptote: $y = -5$

4. Graph Behavior:
   To understand the behavior, we note that:

   • For $x \to -4^{+}$, $\frac{5}{x+4} \to +\infty$, so $y \to +\infty$.
   • For $x \to -4^{-}$, $\frac{5}{x+4} \to -\infty$, so $y \to -\infty$.

This indicates that the function will have two branches: one in which $y$ rises steeply towards $+\infty$ as $x \to -4^{+}$, and one in which $y$ drops steeply towards $-\infty$ as $x \to -4^{-}$.

Would you like to see the graph of this function, or do you have any other specific questions about this function?

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Here are 5 related questions for further exploration:

1. How would you find the intercepts of this function?
2. What is the behavior of the function as $x \to -4$ from the left and right?
3. How does the graph shift when the constant term outside the fraction is altered?
4. What are some real-world applications of rational functions like this one?
5. How do vertical and horizontal asymptotes affect the domain and range of a function?

Tip: When graphing rational functions, always start by identifying the asymptotes, as they give a clear structure to the overall shape and behavior of the graph.