The given function is:

$f(x) = \frac{(x - 1)(x + 1)}{x + 2}$

Analysis of the Function:

1. Zeros of the Function:
   The numerator is $(x - 1)(x + 1)$. The zeros occur when the numerator is zero: $x - 1 = 0 \quad \Rightarrow \quad x = 1$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$

2. Vertical Asymptotes:
   The denominator is $x + 2$. A vertical asymptote occurs when the denominator is zero: $x + 2 = 0 \quad \Rightarrow \quad x = -2$

3. Behavior Near the Vertical Asymptote: As $x$ approaches $-2$, $f(x)$ grows towards positive or negative infinity, depending on the direction.

4. Horizontal Asymptote:
   For large $x$ values, the numerator $(x - 1)(x + 1) \approx x^2$ and the denominator $x + 2 \approx x$. The leading term in the fraction is: $\frac{x^2}{x} = x$ Hence, the function grows without bound. There is no horizontal asymptote, but the function has a slant asymptote.

5. Factoring and Simplification: The given function simplifies to: $f(x) = x - 1 + \frac{3}{x + 2}$ This reveals the slant asymptote: $y = x - 1$

Key Features of the Graph:

1. Zeros: $x = 1$ and $x = -1$.
2. Vertical Asymptote: $x = -2$.
3. Slant Asymptote: $y = x - 1$.
4. Behavior near asymptotes and at infinity:
   • As $x \to \infty$, $f(x) \to x - 1$.
   • As $x \to -\infty$, $f(x) \to x - 1$.

Would you like me to plot the graph for you or provide more details about analyzing it?

Relative Questions:

1. How can the vertical asymptote of $f(x)$ affect the graph’s continuity?
2. What are slant asymptotes, and how do they differ from horizontal asymptotes?
3. Why is the numerator important in determining the zeros of the function?
4. What happens to the graph when $x$ approaches the zeros or vertical asymptote?
5. Can the behavior at infinity always be used to identify the asymptotes?

Tip:

Always simplify the function to reveal hidden asymptotes or patterns—it helps in analyzing and sketching the graph more effectively.