https://r2.erweima.ai/mathbot/image/1725902111778bceka5ko.jpg

Let's analyze the two mathematical problems in the image:

### Problem 3:
**Function:**  
\[
y = \frac{3 + 2x}{x - 6}
\]
**Tasks:**
- Determine the domain.
- Find the image (range).
- Identify the intercept with the origin.
- Determine the asymptotes.
- Draw the graph of the function.

### Problem 4:
**Function:**  
\[
y = -2|x+1| + 4
\]
**Tasks:**
- Determine the vertex of the function.
- Find the intersections with the axes.
- Determine the set of positivity (where the function is positive).
- Determine the set of negativity (where the function is negative).
- Draw the graph of the function.

I will solve each problem step-by-step. Let's start with **Problem 3**:

#### Problem 3: Solution
1. **Domain:**
   The function is undefined where the denominator is zero:  
   \( x - 6 = 0 \implies x = 6 \).  
   Therefore, the domain is all real numbers except \( x = 6 \):  
   \[ \text{Domain: } \mathbb{R} \setminus \{6\} \]

2. **Image (Range):**
   Since the function is a rational function, it can take any real value except where the denominator would make the function undefined. In this case, there are no further restrictions on the range, so it can be all real numbers except where the horizontal asymptote is found.

3. **Intercept with the Origin:**
   - **X-intercept:** Set \( y = 0 \):  
   \[
   0 = \frac{3 + 2x}{x - 6} \implies 3 + 2x = 0 \implies x = -\frac{3}{2}
   \]
   - **Y-intercept:** Set \( x = 0 \):  
   \[
   y = \frac{3 + 2 \cdot 0}{0 - 6} = \frac{3}{-6} = -\frac{1}{2}
   \]

   So, the intercepts are at \( (-\frac{3}{2}, 0) \) and \( (0, -\frac{1}{2}) \).

4. **Asymptotes:**
   - **Vertical Asymptote:** The vertical asymptote occurs where the denominator is zero, i.e., at \( x = 6 \).
   - **Horizontal Asymptote:** Since the degree of the numerator and the denominator is the same (both are linear), the horizontal asymptote is given by the ratio of the leading coefficients:  
   \[
   y = \frac{2}{1} = 2
   \]

5. **Graph:**
   To graph the function, plot the intercepts, the vertical asymptote at \( x = 6 \), and the horizontal asymptote at \( y = 2 \). Then, determine the behavior of the function near the asymptotes.

#### Problem 4: Solution
1. **Vertex:**
   The function is of the form \( y = a|x - h| + k \), where \( (h, k) \) is the vertex.  
   For \( y = -2|x + 1| + 4 \), rewrite it as \( y = -2|x - (-1)| + 4 \).  
   The vertex is \( (-1, 4) \).

2. **Intersections with the Axes:**
   - **X-intercepts:** Set \( y = 0 \):  
   \[
   0 = -2|x+1| + 4 \implies |x+1| = 2 \implies x+1 = 2 \text{ or } x+1 = -2
   \]
   Solving these, we get \( x = 1 \) and \( x = -3 \).  
   So, the x-intercepts are \( (1, 0) \) and \( (-3, 0) \).
   - **Y-intercept:** Set \( x = 0 \):  
   \[
   y = -2|0+1| + 4 = -2 \cdot 1 + 4 = 2
   \]
   So, the y-intercept is \( (0, 2) \).

3. **Positivity and Negativity:**
   - **Positivity:** The function is positive where \( y > 0 \). Solving for \( y > 0 \):  
   \[
   -2|x+1| + 4 > 0 \implies |x+1| < 2 \implies -2 < x+1 < 2 \implies -3 < x < 1
   \]
   So, the function is positive in the interval \( (-3, 1) \).

   - **Negativity:** The function is negative where \( y < 0 \). Solving for \( y < 0 \):  
   \[
   |x+1| > 2 \implies x+1 > 2 \text{ or } x+1 < -2
   \]
   This gives \( x > 1 \) or \( x < -3 \).  
   So, the function is negative in the intervals \( (-\infty, -3) \cup (1, \infty) \).

4. **Graph:**
   To graph the function, plot the vertex at \( (-1, 4) \), the intercepts, and the intervals where the function is positive and negative.

Would you like me to provide a more detailed explanation or visualize the graphs? 

### Follow-up Questions:
1. How do we find the domain of more complex rational functions?
2. What are the steps to determine asymptotes for various types of functions?
3. How do absolute value functions affect the graph and its properties?
4. Can you explain how to find positivity and negativity sets for quadratic functions?
5. What is the importance of intercepts and asymptotes in analyzing a graph?

### Tip:
For rational functions, always check the degrees of the numerator and denominator to determine horizontal asymptotes.

3. Graficá la siguiente función, determinando su dominio, imagen, raíz ordenada al origen y asíntotas: y = (3 + 2x)/(x - 6)
4. Graficá la función y = -2|x+1| + 4 , determinando su vértice, intersecciones con los ejes, conjunto de positividad y de negatividad.

This problem involves graphing a rational function y = (3 + 2x)/(x - 6) and an absolute value function y = -2|x+1| + 4. It covers topics such as determining the domain, asymptotes, intercepts, vertex, and the intervals of positivity and negativity for students in Grades 9-12.

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