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The graph you provided appears to show a function that exhibits asymptotic behavior along the y-axis and a rapid increase for both positive and negative values of \( x \). This type of behavior is characteristic of a logarithmic or reciprocal function.

Given the steep rise on both sides of the graph and its vertical asymptote at \( x = 0 \), this graph likely represents a **reciprocal function**, of the form:

\[
f(x) = \frac{1}{x}
\]

Here's why:
1. **Vertical Asymptote**: There is a vertical asymptote along the y-axis, which corresponds to \( x = 0 \), where the function is undefined.
2. **Behavior for Positive and Negative \( x \)**: For \( x > 0 \), the function increases steeply, and for \( x < 0 \), it decreases steeply in the opposite direction, which is typical of reciprocal functions.

Would you like more detailed analysis or to confirm this? Here are five questions to deepen your understanding:

1. How does the vertical asymptote of a function relate to its domain?
2. What are the horizontal asymptotes of the function \( f(x) = \frac{1}{x} \)?
3. How can you identify asymptotic behavior in a graph?
4. What transformations could be applied to the basic function \( f(x) = \frac{1}{x} \) to shift or stretch it?
5. How does the function behave as \( x \) approaches \( 0 \) from both sides?

**Tip**: Reciprocal functions have undefined values at their vertical asymptotes—make sure to check the domain carefully!

Explore the characteristics of the reciprocal function f(x) = 1/x, with a focus on its vertical asymptote and behavior for positive and negative values of x. This problem is suitable for high school students learning about functions and asymptotes.

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