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?

---

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.