The expression you provided, $\frac{2}{x+5}$, is a rational function. Here’s an analysis of this function:

Key Points:

1. Domain: The expression is undefined where the denominator is zero. So, set $x + 5 = 0$ to find that $x = -5$ is not in the domain. The domain is all real numbers except $x = -5$.

2. Vertical Asymptote: At $x = -5$, the function has a vertical asymptote, meaning the function approaches infinity or negative infinity as $x$ gets closer to $-5$.

3. Horizontal Asymptote: Since the degree of the polynomial in the numerator (0, since it’s a constant) is less than the degree of the polynomial in the denominator (1, because of $x + 5$), the horizontal asymptote is $y = 0$.

4. Graph Behavior: The function will approach the asymptotes but never touch them. For $x > -5$, the function remains positive, and for $x < -5$, it is also positive. The function is symmetric about the vertical asymptote line at $x = -5$.

5. Value at Specific Points: You can substitute values of $x$ to get specific values for the function, like:

   • $x = 0$: $\frac{2}{0+5} = \frac{2}{5}$
   • $x = -4$: $\frac{2}{-4+5} = 2$

Would you like a graphical representation or further explanation of any specific part of this analysis?

Here are five follow-up questions you might consider:

1. What happens if you add or subtract a constant outside of the fraction?
2. How does multiplying the fraction by another number affect the vertical and horizontal asymptotes?
3. What if we instead had $\frac{2}{x^2 + 5}$? How would the properties change?
4. Can we rewrite this function in a different form that makes asymptotes or domain restrictions clearer?
5. How does this function behave as $x$ approaches positive or negative infinity?

Tip: Understanding asymptotes is crucial when analyzing rational functions, as they reveal the behavior of the function at extreme values of $x$ and at points where the function is undefined.