f(x) = (x² - 9)/(x² - 5x + 6). Identify and clearly show all vertical asymptotes, any holes in the graph, horizontal asymptotes, and x-intercepts.

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Let's analyze the rational function f(x) equals x squared minus 9 divided by x squared minus 5x plus 6. First, we'll factor both the numerator and denominator. The numerator x squared minus 9 factors as (x minus 3) times (x plus 3). The denominator x squared minus 5x plus 6 factors as (x minus 2) times (x minus 3). Notice that (x minus 3) is a common factor in both the numerator and denominator. After canceling this common factor, we get the simplified function f(x) equals (x plus 3) divided by (x minus 2), for x not equal to 3. Now let's identify the vertical asymptote and hole in our function. The vertical asymptote occurs at x equals 2, where the denominator of our simplified function equals zero. As x approaches 2, the function values approach positive or negative infinity. Next, we have a hole at x equals 3. This occurs because x minus 3 was a common factor that we canceled from both the numerator and denominator. To find the y-coordinate of this hole, we substitute x equals 3 into our simplified function. This gives us 3 plus 3 divided by 3 minus 2, which equals 6 divided by 1, or simply 6. Therefore, we have a hole at the point (3, 6). Now let's examine the horizontal asymptote and x-intercept of our function. To find the horizontal asymptote, we look at the behavior of the function as x approaches infinity. Since both the numerator and denominator have the same degree, which is 2, the horizontal asymptote is determined by the ratio of the leading coefficients. Both leading coefficients are 1, so the horizontal asymptote is y equals 1. For the x-intercept, we need to find where the numerator equals zero. From the factored form (x minus 3)(x plus 3) equals 0, we get x equals negative 3 or x equals 3. However, since x equals 3 creates a hole rather than an intercept, the only x-intercept is at the point negative 3, 0. Let's summarize all the key features of our rational function. After factoring, we found that f(x) equals (x minus 3)(x plus 3) divided by (x minus 2)(x minus 3), which simplifies to (x plus 3) divided by (x minus 2) for x not equal to 3. The function has a vertical asymptote at x equals 2, where the denominator of the simplified function equals zero. There's a horizontal asymptote at y equals 1, determined by the ratio of the leading coefficients. We identified a hole at the point (3, 6), which occurs because x minus 3 was a common factor in both the numerator and denominator. The function has an x-intercept at negative 3, 0, where the numerator equals zero. The domain of the function includes all real numbers except x equals 2 and x equals 3. As x approaches 2 from the right, the function approaches positive infinity, and as x approaches 2 from the left, the function approaches negative infinity. As x approaches infinity in either direction, the function approaches the horizontal asymptote y equals 1. To summarize our analysis of the rational function f(x) equals x squared minus 9 divided by x squared minus 5x plus 6: We identified a vertical asymptote at x equals 2, where the denominator of the simplified function equals zero. We found a hole at the point (3, 6), which resulted from canceling the common factor x minus 3 from both the numerator and denominator. The function has a horizontal asymptote at y equals 1, determined by the ratio of the leading coefficients of the numerator and denominator. We located an x-intercept at negative 3, 0, where the numerator equals zero. And finally, we determined that the domain of the function includes all real numbers except x equals 2 and x equals 3. When analyzing any rational function, it's essential to factor both the numerator and denominator to identify these key features.