https://file.aiquickdraw.com/mathbot/file/1726265195159bn9y725q.jpg

The graph you've provided shows a function that appears to be an inverse variation or reciprocal function, resembling \( y = \frac{1}{x} \). Based on the curve, the function's behavior is as follows:

- As \( x \) approaches 0 from the negative side, \( y \) increases without bound.
- As \( x \) approaches 0 from the positive side, \( y \) decreases without bound.
- The function does not touch the \( y \)-axis (vertical asymptote at \( x = 0 \)), and the curve extends infinitely upward and downward.

### Range:
The range of the function is all real numbers except \( y = 0 \), because the curve never crosses the x-axis.

Thus, the range is:
\[ (-\infty, 0) \cup (0, \infty) \]

Would you like more details or have any questions?

Here are 5 related questions you might find useful:
1. How do we determine the domain of a reciprocal function?
2. What does an asymptote signify in a function's graph?
3. Can you explain why \( y = 0 \) is excluded from the range of the function?
4. How would the graph change if the function were \( y = \frac{1}{x-2} \)?
5. What are the key differences between a vertical and horizontal asymptote?

**Tip:** For functions with vertical and horizontal asymptotes, always check how the function behaves as it approaches those asymptotes to determine its domain and range.

What is the range of the function shown in the graph?

Explore the range of a reciprocal function with vertical asymptotes using the example of y = 1/x. This problem involves understanding how the function behaves as x approaches 0 and why y = 0 is excluded from the range. Suitable for high school students.

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