Problem 25 - 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.) $$ f(x)=\frac{x^{2}-2}{x^{2}-4} $$

Step-by-Step Solution

Answer

The horizontal asymptote is \(y = 1\), and the vertical asymptotes are \(x = 2\) and \(x = -2\).

Step 1: Identify degrees of numerator and denominator

The degree of the numerator \(x^2 - 2\) is 2, and the degree of the denominator \(x^2 - 4\) is also 2. Since the degrees are equal, we can proceed to finding the horizontal asymptote.

Step 2: Find horizontal asymptote

When the degrees of the numerator and denominator are equal, the horizontal asymptote can be found by dividing the leading coefficients. In this case, the leading coefficients for both the numerator and denominator are 1. Thus, the horizontal asymptote is: $$ y = \frac{1}{1} = 1 $$

Step 3: Find vertical asymptotes

To find the vertical asymptotes, we set the denominator equal to zero and solve for x: $$ x^2 - 4 = 0 $$ Factoring the quadratic equation, we get: $$ (x - 2)(x + 2) = 0 $$ This yields two possible solutions for x: $$ x - 2 = 0 \implies x = 2 \\ x + 2 = 0 \implies x = -2 $$ So, there are two vertical asymptotes at \(x = 2\) and \(x = -2\). In summary, the function \(f(x) = \frac{x^2 - 2}{x^2 - 4}\) has horizontal and vertical asymptotes at the following: - Horizontal Asymptote: \(y = 1\) - Vertical Asymptotes: \(x = 2\) and \(x = -2\)

Key Concepts

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches as the input (or x-value) goes towards positive or negative infinity. In the context of rational functions, finding the horizontal asymptote involves checking the degrees of the function's numerator and denominator.
In this particular exercise, the degrees of both the numerator \(x^2 - 2\) and the denominator \(x^2 - 4\) are 2. When the degrees are equal, the horizontal asymptote can be determined by dividing the leading coefficients.

Here, the leading coefficient of both the numerator and denominator is 1. Thus, the horizontal asymptote is found as follows:

• Divide the leading coefficient of the numerator (1) by the leading coefficient of the denominator (1).
• This results in the equation \(y = \frac{1}{1} = 1\).

Therefore, the horizontal asymptote for this function is the line \(y = 1\). The graph will approach this line as x becomes very large or very small, but it will never actually touch it.

Vertical Asymptote

Vertical asymptotes are vertical lines that a function approaches but never crosses or touches. For rational functions, vertical asymptotes are found where the denominator is zero and the numerator is not zero.
In the exercise given, the denominator of the function is \(x^2 - 4\). To find the vertical asymptotes, we set this equation to zero:
\[x^2 - 4 = 0\]

Factoring this expression gives \((x - 2)(x + 2) = 0\). Solving for x gives two roots:

• \(x - 2 = 0\), leading to \(x = 2\)
• \(x + 2 = 0\), leading to \(x = -2\)

These values of x are where the vertical asymptotes occur. They are \(x = 2\) and \(x = -2\). The function will approach these lines but will not cross them, creating a clear division in the graph's structure at these points.

Rational Functions

Rational functions are a special type of function formed by the ratio of two polynomial expressions. That means they look like \(rac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
Understanding rational functions is crucial because they have distinctive features like asymptotes due to their inherent structure. As a general rule:

• The degree of the numerator and denominator determines the horizontal asymptote.
• Values that make the denominator zero but not the numerator provide the vertical asymptotes.

These functions often have restricted domains, as the values that make the denominator zero are not valid inputs. In this problem, the rational function \(f(x) = \frac{x^2 - 2}{x^2 - 4}\) has these classic characteristics. Recognizing the numerator and denominator aids in determining asymptotic behavior, leading to a clearer graph and understanding of behavior near asymptotes.