The function you provided is:

$f(x) = \frac{1}{x^2 + 2x - 3}$

Steps to Analyze the Function:

1. Factor the denominator: The denominator is a quadratic expression $x^2 + 2x - 3$, which can be factored as: $x^2 + 2x - 3 = (x + 3)(x - 1)$ Thus, the function can be rewritten as: $f(x) = \frac{1}{(x + 3)(x - 1)}$

2. Domain of the function: The domain is all real values of $x$ except where the denominator is zero. To find these points: $(x + 3)(x - 1) = 0$ Solving for $x$, we get $x = -3$ and $x = 1$. Therefore, the domain is: $x \in (-\infty, -3) \cup (-3, 1) \cup (1, \infty)$

3. Asymptotes:

   • Vertical Asymptotes: Occur at $x = -3$ and $x = 1$, where the denominator becomes zero.
   • Horizontal Asymptote: Since the degree of the numerator (constant) is less than the degree of the denominator (quadratic), the horizontal asymptote is at $y = 0$.

4. Behavior around the asymptotes: The function will approach positive or negative infinity as $x$ gets close to $-3$ and $1$, depending on the direction of approach.

Would you like further details on how the function behaves in specific intervals?

Related Questions:

1. How do you determine the domain of a function involving a rational expression?
2. What are the steps to factor a quadratic equation?
3. How do you find vertical and horizontal asymptotes of rational functions?
4. How do you analyze the end behavior of rational functions?
5. Can you explain how to graph a rational function?

Tip: Always check for points of discontinuity when dealing with rational functions.