Graph f(x) = (2x + 3)/(x - 1), clearly show and find vertical and horizontal asymptotes.

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Let's analyze the rational function f of x equals 2x plus 3 divided by x minus 1. We need to find its vertical and horizontal asymptotes. First, let's graph the function. Notice that the function has two separate branches because it's undefined at x equals 1. This creates a vertical asymptote at x equals 1, shown by the red dashed line. As x approaches 1 from the left, the function approaches negative infinity. As x approaches 1 from the right, the function approaches positive infinity. For the horizontal asymptote, we look at the behavior as x approaches infinity. The ratio of the leading coefficients is 2 divided by 1, which equals 2. So the horizontal asymptote is y equals 2, shown by the green dashed line. Now let's focus on finding the vertical asymptote of our function. A vertical asymptote occurs where the denominator equals zero but the numerator doesn't. For our function, we set the denominator x minus 1 equal to zero, which gives us x equals 1. Next, we check if the numerator is non-zero at this point. When x equals 1, the numerator equals 2 times 1 plus 3, which is 5. Since 5 is not zero, x equals 1 is indeed a vertical asymptote. Looking at the graph, we can see that as x approaches 1 from the left, the function values decrease without bound, approaching negative infinity. And as x approaches 1 from the right, the function values increase without bound, approaching positive infinity. This confirms our algebraic finding that x equals 1 is a vertical asymptote. Now let's find the horizontal asymptote of our function. A horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity. To find it, we first compare the degrees of the numerator and denominator. In our function, both the numerator 2x plus 3 and the denominator x minus 1 have degree 1. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. So the horizontal asymptote is y equals 2. Looking at our graph with a wider x-range, we can confirm this result. As x approaches negative infinity, the function values approach 2 from below. And as x approaches positive infinity, the function values approach 2 from above. This verifies that y equals 2 is indeed the horizontal asymptote. Now let's analyze the complete graph of our function f of x equals 2x plus 3 divided by x minus 1. First, let's note that the domain of this function is all real numbers except x equals 1, where the function is undefined. We've identified the vertical asymptote at x equals 1 and the horizontal asymptote at y equals 2. The function consists of two separate branches. The left branch exists for all x less than 1. As x approaches 1 from the left, the function values decrease without bound, approaching negative infinity. The right branch exists for all x greater than 1. As x approaches 1 from the right, the function values increase without bound, approaching positive infinity. Both branches approach the horizontal asymptote y equals 2 as x approaches positive or negative infinity. Let's highlight some key points on the graph. At x equals 0, the function value is negative 3. At x equals negative 1, the function value is 0.5. At x equals 2, the function value is 7. And at x equals 3, the function value is 4.5. Notice how the function decreases on the left branch and increases then decreases on the right branch. Let's summarize what we've learned about the rational function f of x equals 2x plus 3 divided by x minus 1. First, we identified that this function has a vertical asymptote at x equals 1, where the denominator equals zero. This creates a discontinuity in the graph, dividing it into two separate branches. Second, we found that the function has a horizontal asymptote at y equals 2, which is the ratio of the leading coefficients of the numerator and denominator. The domain of this function is all real numbers except x equals 1, where the function is undefined. The function consists of two branches: the left branch for x less than 1, which decreases without bound as x approaches 1 from the left; and the right branch for x greater than 1, which decreases from positive infinity as x approaches 1 from the right. Both branches approach the horizontal asymptote y equals 2 as x approaches positive or negative infinity. Understanding asymptotes is crucial for analyzing the behavior of rational functions, especially near points where they're undefined and as the input approaches infinity.