$\frac{-3(x-2)(x+7)}{x^2+5 x}$

To analyze the function $f(x) = \frac{-3(x-2)(x+7)}{x^2+5x}$, we first simplify the expressions. The numerator is $N(x) = -3(x^2 + 5x - 14) = -3x^2 - 15x + 42$. The denominator is $D(x) = x^2 + 5x$. The function is defined for all real numbers where the denominator is not zero. Setting $D(x) = x(x+5) = 0$, we find the domain of the function is all real numbers except $x = 0$ and $x = -5$.

Vertical Asymptotes

The vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Since $D(x) = 0$ at $x = 0$ and $x = -5$, and the numerator $N(0) = -3(-2)(7) = 42 eq 0$ and $N(-5) = -3(-7)(2) = 42 eq 0$, the vertical asymptotes are the lines $x = 0$ and $x = -5$.

Horizontal Asymptote

To find the horizontal asymptote, we examine the behavior of $f(x)$ as $x \to \pm \infty$. Since both the numerator and denominator are quadratic polynomials (degree 2), we look at the ratio of their leading coefficients. The leading coefficient of the numerator is $-3$, and the leading coefficient of the denominator is $1$. Therefore, the horizontal asymptote is $y = \frac{-3}{1} = -3$.

Intercepts and Summary

The $x$-intercepts are the values of $x$ where $f(x) = 0$, which occurs when the numerator is zero. Solving $-3(x-2)(x+7) = 0$, we find the $x$-intercepts at $x = 2$ and $x = -7$. Since $x = 0$ is not in the domain of the function, there is no $y$-intercept. The final characteristics are: Domain $\{x \in \mathbb{R} \mid x eq 0, -5\}$, vertical asymptotes $x = 0, x = -5$, horizontal asymptote $y = -3$, and $x$-intercepts $x = 2, x = -7$.

Examples

Rational functions appear in many real-world applications. For instance, in economics, a cost-per-unit function is often represented as a rational function: Total Cost divided by the quantity produced. If fixed costs are present, as production increases, the fixed cost per unit decreases, approaching a horizontal asymptote representing the variable cost per unit. Understanding asymptotes and domain limitations allows economists to model production efficiency and identify capacity constraints.