Given that 𝑓(𝑥) = 1/𝑥²−1 , sketch the graph of 𝑦 = 𝑓(𝑥).

To sketch the graph of the function \( y = f(x) = \frac{1}{x^2 - 1} \), let's analyze its key features step by step:

### 1. Domain
The function is undefined where the denominator is zero, so let's find those points:

\[
x^2 - 1 = 0 \implies x = \pm 1
\]

Thus, the domain of the function is all real numbers except \( x = \pm 1 \).

### 2. Asymptotes
- **Vertical Asymptotes**: Occur where the function is undefined:
  - \( x = 1 \)
  - \( x = -1 \)
  
- **Horizontal Asymptote**: As \( x \to \infty \) or \( x \to -\infty \), the function approaches zero because the numerator is constant and the denominator grows without bound:
  - Horizontal asymptote: \( y = 0 \).

### 3. Intercepts
- **x-intercepts**: The function has no x-intercepts because the numerator is always 1, never zero.
- **y-intercept**: 
  - Set \( x = 0 \):
  
  \[
  y = f(0) = \frac{1}{0^2 - 1} = -1
  \]
  
  So, the y-intercept is \( (0, -1) \).

### 4. Behavior Near Asymptotes
- As \( x \to 1^+ \) or \( x \to 1^- \), \( f(x) \to \infty \).
- As \( x \to -1^+ \) or \( x \to -1^- \), \( f(x) \to \infty \).

### 5. Sign of the Function
- For \( x > 1 \), \( x^2 - 1 > 0 \), so \( f(x) > 0 \).
- For \( -1 < x < 1 \), \( x^2 - 1 < 0 \), so \( f(x) < 0 \).
- For \( x < -1 \), \( x^2 - 1 > 0 \), so \( f(x) > 0 \).

### 6. Sketch of the Graph
- The graph has two vertical asymptotes at \( x = 1 \) and \( x = -1 \).
- There is a horizontal asymptote at \( y = 0 \).
- The graph passes through the point \( (0, -1) \).

The graph will have:
- Two branches in the regions \( x < -1 \) and \( x > 1 \) approaching the asymptotes.
- A section between \( -1 < x < 1 \) where the function is negative and approaches negative infinity near the vertical asymptotes.

### Summary of the Graph
- Vertical asymptotes at \( x = 1 \) and \( x = -1 \).
- Horizontal asymptote at \( y = 0 \).
- Decreasing and approaching infinity near vertical asymptotes.
- The function is negative between \( -1 \) and \( 1 \).

Would you like a graphical representation of this description, or do you have any specific questions?

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**Related Questions:**

1. How do we determine asymptotes for more complex rational functions?
2. What changes in the graph if \( f(x) = \frac{1}{x^2 + 1} \)?
3. How can we find points of inflection for rational functions?
4. What effect does a shift \( f(x) = \frac{1}{x^2 - 1} + 2 \) have on the graph?
5. How does multiplying the function by a constant affect its graph?

**Tip:** When dealing with rational functions, always identify the vertical and horizontal asymptotes first as they help in sketching the general shape of the graph.

Learn how to sketch the graph of the function y = 1/(x^2 - 1) with detailed steps and analysis of its domain, asymptotes, intercepts, and behavior near asymptotes. This problem covers key concepts in rational functions and asymptotic behavior, suitable for high school students.

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