Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{3 x}{x^{2}-x-6} $$

Step-by-Step Solution

Answer

In summary, the vertical asymptotes of the function \(f(x) = \frac{3x}{x^2 - x - 6}\) are at \(x = 3\) and \(x = -2\), and the horizontal asymptote is at \(y = 0\).

Step 1: Identify the given function

The given function is \(f(x) = \frac{3x}{x^2 - x - 6}\).

Step 2: Find the vertical asymptotes

To find the vertical asymptotes, we must set the denominator of the function equal to zero and solve for the x values. This gives us: \(x^2 - x - 6 = 0\) Now, to find the roots, we'll factor the quadratic equation as follows: \((x - 3)(x + 2) = 0\) The solutions for the equation are: \(x = 3\) and \(x = -2\) These values represent the vertical asymptotes of the function.

Step 3: Find the horizontal asymptote.

To find the horizontal asymptote, we need to analyze the behavior of the function as x approaches infinity. We can compare the degree of the numerator and the degree of the denominator to determine the horizontal asymptote. The degree of the numerator of our function is 1, and the degree of the denominator is 2. When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at \(y=0\). In conclusion: - Vertical asymptotes: \(x = 3\) and \(x = -2\) - Horizontal asymptote: \(y = 0\)

Key Concepts

Vertical Asymptotes

Understanding vertical asymptotes is crucial when studying graphs of functions. A vertical asymptote occurs where the function's value becomes infinitely large as it approaches a certain value of x. Essential in rational functions, like the provided exercise

Identifying Vertical Asymptotes

To find vertical asymptotes, we zero the denominator and solve for x. For function
\(f(x) = \frac{3x}{x^2 - x - 6}\), factoring gives us \((x - 3)(x + 2) = 0\). The resulting x-values, 3 and -2, are where the function cannot exist, as it would require division by zero.

• For the function \(f(x) = \frac{3x}{x^2 - x - 6}\), we establish vertical asymptotes at \(x = 3\) and \(x = -2\).
• These lines on the graph of the function indicate where the function's value heads towards infinity.

When graphing, these asymptotes are drawn as dashed lines, signifying the function’s approach but never actually touching these lines.

Horizontal Asymptote

Unlike vertical asymptotes, a horizontal asymptote reflects the behavior of a function as x goes toward infinity or negative infinity. It's a horizontal line that the function approaches but does not necessarily reach or cross.

Determining the Horizontal Asymptote

In algebra, the degree (the highest exponent) of the numerator and denominator dictates the position of the horizontal asymptote. If the numerator's degree is less than the denominator's, as x increases or decreases without bound, the function approaches zero. That means the horizontal asymptote is at \(y=0\) for \(f(x) = \frac{3x}{x^2 - x - 6}\).

• For our function, since the denominator's degree is greater, the horizontal asymptote occurs at \(y=0\).
• This line represents the long-term behavior of the function far along the x-axis.

As x grows large in magnitude, the impact of the function’s value diminishes, flattening out towards the asymptote, without crossing it.

Rational Functions

A rational function is a ratio of two polynomials. It is expressed as \(f(x) = \frac{P(x)}{Q(x)}\), where both P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial. The values which make Q(x) equal to zero are not part of the function's domain.

Characteristics of Rational Functions

Rational functions exhibit various key features, such as vertical and horizontal asymptotes, intercepts, and possibly oblique (slant) asymptotes, depending on the degrees of the polynomials.
Identifying these aspects is paramount for understanding the function's graph and behavior.

• Rational functions can be simplified by factoring and reducing common terms, but this does not remove the asymptotes.
• The vertical asymptotes are found at values of x for which the denominator Q(x) is zero, as seen with \(x = 3\) and \(x = -2\).
• The horizontal asymptote is determined by comparing the degrees of the numerator and denominator, indicative of the function's end behavior.

In essence, rational functions offer insights into limits and continuity within calculus and are often used to model real-world scenarios where proportions are essential.