$\frac{x^2-2 x-8}{3 x-2}$

To analyze the rational function $f(x) = \frac{x^2 - 2x - 8}{3x - 2}$, we first factor the numerator. The quadratic expression $x^2 - 2x - 8$ can be factored by finding two numbers that multiply to $-8$ and add to $-2$. These numbers are $-4$ and $2$, so the numerator is $(x - 4)(x + 2)$. Thus, the function is $f(x) = \frac{(x - 4)(x + 2)}{3x - 2}$.

Determining Domain and Vertical Asymptote

The expression is defined for all real numbers except where the denominator is zero. Setting the denominator equal to zero, we have $3x - 2 = 0$, which gives $x = \frac{2}{3}$. Therefore, the domain of the function is all real numbers $x eq \frac{2}{3}$. Since the numerator is not zero at $x = \frac{2}{3}$, there is a vertical asymptote at $x = \frac{2}{3}$.

Finding Intercepts

To find the intercepts, we check for values where the numerator or denominator are zero. The x-intercepts occur where the numerator is zero: $x - 4 = 0$ or $x + 2 = 0$, which gives $x = 4$ and $x = -2$. The y-intercept occurs when $x = 0$, calculated as $f(0) = \frac{0^2 - 2(0) - 8}{3(0) - 2} = \frac{-8}{-2} = 4$. The point $(0, 4)$ is the y-intercept.

Identifying Slant Asymptote

To determine the end behavior and the slant asymptote, we use polynomial long division to divide $x^2 - 2x - 8$ by $3x - 2$. Dividing $x^2$ by $3x$ gives $\frac{1}{3}x$. Multiplying $\frac{1}{3}x(3x - 2) = x^2 - \frac{2}{3}x$. Subtracting this from $x^2 - 2x - 8$ leaves $-\frac{4}{3}x - 8$. Next, dividing $-\frac{4}{3}x$ by $3x$ gives $-\frac{4}{9}$. Multiplying $-\frac{4}{9}(3x - 2) = -\frac{4}{3}x + \frac{8}{9}$. Subtracting this from $-\frac{4}{3}x - 8$ yields a remainder of $-\frac{80}{9}$. Therefore, the function can be written as $f(x) = \frac{1}{3}x - \frac{4}{9} - \frac{80/9}{3x-2}$. The slant asymptote is the line $y = \frac{1}{3}x - \frac{4}{9}$.

Examples

Understanding rational functions and their asymptotes is essential in engineering and economics. For example, if a company's average cost per unit is modeled by a rational function, the horizontal or slant asymptotes reveal the long-term minimum average cost as production increases. Identifying vertical asymptotes helps engineers avoid critical points where a system might fail or become unstable, such as in electrical circuits or mechanical vibrations.