For the rational function:
$f(x) = \frac{3}{x^2 - 2x - 8}$
Find the following:
To analyze the function \(f(x) = \frac{3}{x^2 - 2x - 8}\), we first factor the denominator to understand its behavior. The quadratic expression \(x^2 - 2x - 8\) can be factored by finding two numbers that multiply to \(-8\) and sum to \(-2\). These numbers are \(-4\) and \(2\), so the expression becomes \((x - 4)(x + 2)\). Thus, the function is \(f(x) = \frac{3}{(x - 4)(x + 2)}\).
The domain of a rational function consists of all real numbers except those that make the denominator zero. Setting the denominator equal to zero, we get \((x - 4)(x + 2) = 0\), which results in \(x = 4\) and \(x = -2\). Since these values make the denominator zero, the function is undefined at these points. Therefore, the domain of \(f(x)\) is all real numbers except \(-2\) and \(4\), expressed in interval notation as \((-\infty, -2) \cup (-2, 4) \cup (4, \infty)\).
To find the zeros of the function, we set the numerator equal to zero. Here, the numerator is the constant \(3\). Since \(3 = 0\) has no solutions, there are no \(x\)-values that make \(f(x) = 0\), meaning there are no zeros. For the \(y\)-intercept, we evaluate the function at \(x = 0\):
\(f(0) = \frac{3}{0^2 - 2(0) - 8} = \frac{3}{-8} = -\frac{3}{8}\).
So, the \(y\)-intercept is at the point \((0, -3/8)\).
Vertical asymptotes occur at the values of \(x\) where the denominator is zero and the numerator is non-zero. As identified earlier, the denominator is zero at \(x = 4\) and \(x = -2\), and at these points, the numerator is \(3\), which is not zero. Thus, the vertical asymptotes are the lines \(x = 4\) and \(x = -2\). Since there are no common factors between the numerator and denominator, there are no holes in the graph.
To find the horizontal asymptote, we analyze the behavior of the function as \(x \to \pm \infty\). Since the degree of the numerator (which is 0) is less than the degree of the denominator (which is 2), the function approaches \(0\) as \(x\) grows arbitrarily large in the positive or negative direction. Therefore, the horizontal asymptote is the line \(y = 0\).
In summary, for the function \(f(x) = \frac{3}{x^2 - 2x - 8}\):
Understanding the characteristics of rational functions, such as asymptotes and intercepts, is essential for modeling real-world phenomena like the concentration of a chemical in a solution over time. For example, if the concentration \(C(t)\) of a substance in a reactor is given by a rational function, determining the horizontal asymptote reveals the steady-state concentration as time \(t\) increases indefinitely. Similarly, identifying vertical asymptotes helps engineers predict critical time points where the system might become unstable or where the concentration values are undefined, ensuring safer and more efficient process control.