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.

Explore the rational function 2/(x+5), including key concepts such as its domain, vertical and horizontal asymptotes, and graph behavior. This analysis is suitable for students in Grades 8-10 and covers essential ideas like undefined points and asymptotic behavior.

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