Problem 92 - College Algebra

Question

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

Step-by-Step Solution

Answer

The simplified equation for \(f\) is \( -\frac{2}{(x+2)(x+3)} \). The graph of the function has vertical asymptotes at \(x = -2\) and \(x = -3\) and a horizontal asymptote at \(y = 0\). The function is negative when \(x < -3\) and \(x > -2\), and positive when \(-3 < x < -2\).

Step 1: Finding the Common Denominator

First, find the common denominator for the two fractions. The denominators are \(x^{2} + 3x + 2\) and \(x^{2} + 4x + 3\). Notice that these are quadratic functions and they can be factored. The factored forms are (x + 1)(x + 2) and (x + 1)(x + 3). Therefore, the common denominator would be (x + 1)(x + 2)(x + 3).

Step 2: Simplifying the Fraction

Now, combine the fractions by subtracting and using the common denominator. When you subtract, remember to distribute the negative sign.\n\( \frac{2\cdot (x+3) - 4\cdot (x+2)}{(x+1)(x+2)(x+3)} = \frac{2x + 6 - 4x -8}{(x+1)(x+2)(x+3)} = \frac{-2x - 2}{(x+1)(x+2)(x+3)} = \frac{-2 \cdot (x + 1)}{(x+1)(x+2)(x+3)} \) Next, cancel common factors to simplify the fraction: \( f(x) = -\frac{2}{(x+2)(x+3)} \)

Step 3: Graphing the function

To graph the function, make note of potential asymptotes, the values for which the denominator of the function will be zero: \(x = -2\) and \(x = -3\). Also, since the degree of the denominator is higher that that of the numerator and there are no common factors, there is a horizontal asymptote at \(y = 0\). Plot these vertical and horizontal asymptotes, indicating their equations. Now, check a few sign changes around the asymptotes, and then plot the final graph, ensuring that the function tends towards the asymptotes as \(x\) goes towards infinity or negative infinity respectively.