Explain and Graph the function f(x) = (3x² + 2x - 5)/(x² - 4)

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Let's analyze the rational function f of x equals 3x squared plus 2x minus 5, divided by x squared minus 4. First, we need to determine the domain. Since division by zero is undefined, we must exclude values where the denominator equals zero. Setting x squared minus 4 equal to zero, we get x equals plus or minus 2. So the domain is all real numbers except x equals 2 and x equals negative 2. These points create vertical asymptotes in our graph, shown by the red dashed lines. Now let's find the horizontal asymptote. Since the degree of the numerator equals the degree of the denominator, both being 2, the horizontal asymptote is the ratio of their leading coefficients. That gives us y equals 3, shown by the green dashed line. Next, we find the intercepts. For x-intercepts, we set the numerator equal to zero and solve 3x squared plus 2x minus 5 equals 0. Factoring gives us 3x plus 5 times x minus 1 equals 0, which yields x equals negative 5 thirds and x equals 1. For the y-intercept, we evaluate the function at x equals 0, giving us f of 0 equals negative 5 divided by negative 4, which equals 5 fourths or 1.25. Let's analyze the behavior of our function in different regions. To the left of x equals negative 2, the function approaches the horizontal asymptote y equals 3 from below as x approaches negative infinity. As x approaches negative 2 from the left, the function increases without bound, approaching positive infinity. In the middle region, between x equals negative 2 and x equals 2, the function approaches negative infinity as x approaches negative 2 from the right. It then passes through the x-intercepts at x equals negative 5 thirds and x equals 1, as well as the y-intercept at y equals 5 fourths. As x approaches 2 from the left, the function decreases without bound, approaching negative infinity. Finally, to the right of x equals 2, the function approaches positive infinity as x approaches 2 from the right, and then decreases, approaching the horizontal asymptote y equals 3 from above as x approaches positive infinity. Let's analyze the algebraic form of our function. We can factor both the numerator and denominator to get a clearer understanding. The numerator 3x squared plus 2x minus 5 factors as 3x plus 5 times x minus 1, while the denominator x squared minus 4 factors as x minus 2 times x plus 2. This factored form helps us identify the x-intercepts at x equals negative 5 thirds and x equals 1, which come from setting the numerator equal to zero. The vertical asymptotes at x equals negative 2 and x equals 2 come from setting the denominator equal to zero. To find the horizontal asymptote, we examine the limit as x approaches infinity. Since the numerator and denominator have the same degree, the limit equals the ratio of their leading coefficients, which is 3. The domain of the function is all real numbers except x equals negative 2 and x equals 2. The range is all real numbers except y equals 3, as the function approaches but never equals its horizontal asymptote. To summarize our analysis of the rational function f of x equals 3x squared plus 2x minus 5, divided by x squared minus 4: First, we identified the vertical asymptotes at x equals negative 2 and x equals 2, where the denominator equals zero. We also found the horizontal asymptote at y equals 3, which is the ratio of the leading coefficients. The function has x-intercepts at x equals negative 5 thirds and x equals 1, and a y-intercept at y equals 5 fourths. The graph consists of three separate branches that never cross the asymptotes. Factoring the function as 3x plus 5 times x minus 1, divided by x minus 2 times x plus 2, helped us identify these key features. Understanding the behavior near asymptotes and the overall shape of each branch is crucial for sketching rational functions accurately. This analysis demonstrates how algebraic manipulation and graphical interpretation work together to provide a complete understanding of rational functions.