Problem 91 - 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. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

Step-by-Step Solution

Answer

The simplified expression for the function \(f(x)\) is \(f(x) = \frac{1}{2} \left(\frac{x^2 - 27}{x^2 - 9}\right)\) for \(x ≠ 0\) and \(x ≠ ±3\). The vertical asymptotes are at \(x = ±3\), the horizontal asymptote is at y = \(\frac{1}{2}\), and the x-intercepts are at \(x = ±\sqrt{27}\).

Step 1: Simplify the operation

We must find a common denominator for the fractions in the expression, then subtract them. We can take \(2x(x^2 - 9)\) as the common denominator. So, we have: \[f(x) = \frac{x(x^2 - 9) - 18x}{2x(x^2-9)} = \frac{x^3 - 9x - 18x}{2x(x^2-9)} = \frac{x^3 - 27x}{2x(x^2-9)}\]

Step 2: Factorize the Polynomial

Now, factor out the common factor of x from the numerator to further simplify the function: \[f(x) = \frac{x(x^2 - 27)}{2x(x^2-9)}.\] For \(x ≠ 0\) and \(x ≠ ±3\), the simplified function is \[f(x) = \frac{1}{2} \left(\frac{x^2 - 27}{x^2 - 9}\right).\]

Step 3: Identify the Asymptotes

The vertical asymptotes of the function are the values of x that make the denominator equal to zero. So, we have two vertical asymptotes, \(x = -3\) and \(x = 3\). Since the degree of the function is the same in the numerator and denominator, the horizontal asymptote can be found by the ratio of leading constants, which is \(\frac{1}{2}\).

Step 4: Find the Intercepts

The x-intercepts are found when the function equals zero. So, set the numerator of the simplified function equal to zero and solve for x: \[x^2 - 27 = 0 \Rightarrow x = ±\sqrt{27}.\] The y-intercept can be found by substituting \(x = 0\) into the equation but it is undefined as \(x = 0\) leads to division by zero in the original function.

Step 5: Graph the function

Create a graph with the horizontal asymptote at y = \(\frac{1}{2}\), vertical asymptotes at \(x = -3\) and \(x = 3\), and the x-intercepts at \(x = +\sqrt{27}\) and \(x = -\sqrt{27}\). Choose several values of x between the vertical asymptotes and beyond to plot additional points and create a smooth curve.