Graph f(x) = (x² - 4)/(x² - 2x - 3) and identify all asymptotes (vertical and horizontal), x-intercepts, and holes in the graph. Show how the function behaves in each region.

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Let's analyze the rational function f(x) equals x squared minus 4 divided by x squared minus 2x minus 3. First, we'll factor both the numerator and denominator. The numerator factors as (x minus 2) times (x plus 2), and the denominator factors as (x minus 3) times (x plus 1). From this factored form, we can identify the key features of the graph. The vertical asymptotes occur where the denominator equals zero, which happens at x equals negative 1 and x equals 3. The horizontal asymptote is at y equals 1, which is the ratio of the leading coefficients. The x-intercepts are at x equals negative 2 and x equals 2, where the numerator equals zero. The y-intercept is at 4/3, which we get by evaluating f(0). Now let's analyze the behavior of the function in different regions. We'll divide the domain into five regions separated by the x-intercepts and vertical asymptotes. In Region 1, from negative infinity to negative 2, the function is positive and decreasing, approaching the horizontal asymptote y equals 1 as x approaches negative infinity. In Region 2, between negative 2 and negative 1, the function is negative and continues to decrease, approaching negative infinity as x approaches the vertical asymptote at negative 1 from the left. In Region 3, between negative 1 and 2, the function starts at positive infinity near x equals negative 1, decreases through the y-intercept at 4/3, crosses the horizontal asymptote at x equals 1/2, and reaches zero at x equals 2. In Region 4, between 2 and 3, the function is negative and decreases toward negative infinity as x approaches 3 from the left. Finally, in Region 5, from 3 to positive infinity, the function starts at positive infinity and decreases, approaching the horizontal asymptote y equals 1 as x approaches positive infinity. Let's take a closer look at the asymptotic behavior of our function. The function has two vertical asymptotes at x equals negative 1 and x equals 3, where the denominator equals zero. As x approaches negative 1 from the left, the function decreases toward negative infinity. As x approaches negative 1 from the right, the function increases from positive infinity. Similarly, as x approaches 3 from the left, the function decreases toward negative infinity, and as x approaches 3 from the right, the function increases from positive infinity. The function also has a horizontal asymptote at y equals 1, which is the ratio of the leading coefficients of the numerator and denominator. As x approaches either positive or negative infinity, the function approaches this horizontal asymptote. Interestingly, the function actually crosses its horizontal asymptote at x equals one-half, which is a characteristic of rational functions where the numerator and denominator have the same degree. Let's analyze the key points of our rational function. The x-intercepts occur where the numerator equals zero. Since the numerator is x squared minus 4, which factors as (x minus 2)(x plus 2), the x-intercepts are at x equals negative 2 and x equals 2. The y-intercept is found by evaluating the function at x equals 0, which gives us negative 4 divided by negative 3, or 4/3. The critical points of the function are x equals negative 2, negative 1, 2, and 3. These points divide the domain into five regions with different behaviors. The points x equals negative 1 and x equals 3 are vertical asymptotes where the function is undefined, while x equals negative 2 and x equals 2 are x-intercepts where the function crosses the x-axis. Looking at the factored form of our function, f(x) equals (x minus 2)(x plus 2) divided by (x minus 3)(x plus 1), we can easily identify these key features and understand how the function behaves in different regions of its domain. Let's summarize the key features of our rational function f(x) equals x squared minus 4 divided by x squared minus 2x minus 3, which can be factored as (x minus 2)(x plus 2) divided by (x minus 3)(x plus 1). First, we identified two vertical asymptotes at x equals negative 1 and x equals 3, where the denominator equals zero and the function is undefined. Second, we found a horizontal asymptote at y equals 1, which the function approaches as x approaches positive or negative infinity. Interestingly, the function crosses this horizontal asymptote at x equals one-half. Third, we determined that the x-intercepts occur at x equals negative 2 and x equals 2, where the numerator equals zero, and the y-intercept is at the point (0, 4/3). Finally, we analyzed how the function's behavior changes at the critical points x equals negative 2, negative 1, 2, and 3, which divide the domain into five distinct regions. Understanding these key features gives us a complete picture of the function's behavior throughout its domain.