https://file.aiquickdraw.com/mathbot/file/1727564754019w1ubipxu.jpg

The function provided in the image is:

\[
g(x) = \frac{3}{x + 2}
\]

This is a rational function. Let me analyze some key properties of this function:

### 1. **Domain:**
   The function is undefined when the denominator equals zero. So, solve for \(x + 2 = 0\):
   \[
   x = -2
   \]
   Therefore, the domain of \(g(x)\) is all real numbers except \(x = -2\):
   \[
   \text{Domain: } (-\infty, -2) \cup (-2, \infty)
   \]

### 2. **Asymptotes:**
   - **Vertical Asymptote:** Since \(g(x)\) becomes undefined when \(x = -2\), there is a vertical asymptote at \(x = -2\).
   - **Horizontal Asymptote:** For large values of \(x\) (positive or negative), the function approaches 0 because the degree of the denominator is greater than the degree of the numerator. Hence, the horizontal asymptote is:
     \[
     y = 0
     \]

### 3. **Intercepts:**
   - **x-intercept:** Set \(g(x) = 0\) and solve:
     \[
     \frac{3}{x + 2} = 0
     \]
     Since 3 can never be 0, there is no x-intercept.
   - **y-intercept:** Set \(x = 0\) and solve for \(g(0)\):
     \[
     g(0) = \frac{3}{0 + 2} = \frac{3}{2}
     \]
     So, the y-intercept is at \( \left(0, \frac{3}{2}\right) \).

Would you like a deeper explanation on any specific part of this function?

Here are 5 follow-up questions related to this:

1. How do you graph a rational function like \( g(x) = \frac{3}{x + 2} \)?
2. What is the behavior of \( g(x) \) near its vertical asymptote \( x = -2 \)?
3. How does the horizontal asymptote \( y = 0 \) influence the behavior of \( g(x) \)?
4. Can we determine any other key points of the function \( g(x) \)?
5. How do transformations like shifts affect rational functions like this?

**Tip:** Always consider the behavior of rational functions both near their asymptotes and at infinity to fully understand their graph and characteristics.

Find the domain, asymptotes, and intercepts of the function g(x) = 3 / (x + 2).

Explore the detailed analysis of the rational function g(x) = 3 / (x + 2), including finding its domain, vertical and horizontal asymptotes, and intercepts. Perfect for students in grades 10-12 studying algebra and rational functions.

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