https://file.aiquickdraw.com/mathbot/file/1728966748349hjvnz37k.jpg

The graph you provided appears to be a rational function that has two vertical asymptotes and one horizontal asymptote. Let's analyze the range of this function based on the visual clues:

### Observations:
1. **Asymptotes:**
   - The vertical asymptotes occur at \( x = 0 \) and \( x = 10 \) (as indicated by the dashed blue lines).
   - The horizontal asymptote appears to be at \( y = 0 \), meaning that the graph approaches but does not touch \( y = 0 \) as \( x \to \pm \infty \).

2. **Behavior near asymptotes:**
   - As \( x \to 0^+ \), \( y \to +\infty \), and as \( x \to 0^- \), \( y \to -\infty \).
   - As \( x \to 10^+ \) and \( x \to 10^- \), the graph also tends toward infinity on both ends.
   
3. **Graph behavior between the vertical asymptotes:**
   - Between \( x = 0 \) and \( x = 10 \), the graph dips below the x-axis, reaching a minimum near \( y = -4 \), and then increases again.

### Range:
- The function takes all values greater than or equal to 0, excluding 0 itself because of the horizontal asymptote.
- Between \( x = 0 \) and \( x = 10 \), the function appears to reach a minimum value of about \( y = -4 \).
  
Therefore, the **range** of the function is:
\[
(-\infty, -4] \cup (0, +\infty)
\]

This means the graph covers all values less than or equal to -4 and all positive values of \( y \), but it never reaches 0.

Would you like more details or explanations?  
Here are five follow-up questions related to this topic:
1. How are vertical asymptotes determined in a rational function?
2. What is the significance of horizontal asymptotes in rational functions?
3. Can a rational function have more than one horizontal asymptote?
4. How can we find the minimum value of the function between two vertical asymptotes?
5. How do horizontal and vertical shifts affect the range of a rational function?

**Tip:** A function's range is determined by its output values, and understanding the behavior near asymptotes is key to finding the range for rational functions.

What is the range of the rational function?

Learn how to determine the range of a rational function using visual clues from a graph. This problem involves analyzing vertical and horizontal asymptotes and understanding how they affect the function's output values. Suitable for students in Grades 10-12.

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