Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x^{2}+2}{x^{2}-16}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{x^{2}+2}{x^{2}-16}\) is all real numbers except \(x = 4\) and \(x = -4\), or in interval notation, \(-\infty, -4\) U \(-4, 4\) U \(4, \infty\). The function has vertical asymptotes at \(x = 4\) and \(x = -4\) and a horizontal asymptote at \(y = 1\).

Step 1: Calculating the Domain

Set the denominator equal to zero and solve for \(x:\)\n\(x^{2}-16 = 0\)\nThis gives \(x = 4\) or \(x = -4\). Therefore, the function is undefined at \(x = 4\) and \(x = -4\). Thus, the domain of the function is all real numbers except \(x = 4\) and \(x = -4\). In interval notation, the domain is:\n \(-\infty, -4\) U \(-4, 4\) U \(4, \infty\)

Step 2: Calculating the Vertical Asymptotes

The values of x for which the function is undefined are also the vertical asymptotes. Therefore, the function has vertical asymptotes at \(x = 4\) and \(x = -4\). These are lines which the function approaches but never crosses.

Step 3: Calculating the Horizontal Asymptote

A horizontal asymptote is a horizontal line that the function approaches as \(x\) approaches positive or negative infinity. We can determine the horizontal asymptote by looking at the degree of the polynomial in the numerator and the degree of the polynomial in the denominator.\nHere, both degrees are 2 (i.e. they have same degree). If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients of the numerator and the denominator.\nIn this case, both coefficients are 1, thus the horizontal asymptote is \(y = 1\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes in a rational function occur at values of \(x\) that make the denominator zero while the numerator is non-zero. These asymptotes represent the values where the function is undefined, causing the graph to shoot up or down sharply, getting infinitely close but never actually touching or crossing the asymptote.
In the case of the function \(f(x) = \frac{x^2 + 2}{x^2 - 16}\), we set the denominator to zero to find the points of vertical asymptotes:

• \(x^2 - 16 = 0\)
• Solves to \(x - 4 = 0\) or \(x + 4 = 0\)
• Thus, the vertical asymptotes are at \(x = 4\) and \(x = -4\)

These lines, \(x = 4\) and \(x = -4\), are the boundaries that the function's graph will approach, but never actually cross.

Horizontal Asymptotes

Horizontal asymptotes occur when a function approaches a particular constant value as \(x\) goes to negative or positive infinity. In rational functions, these are determined by comparing the degrees of the polynomials in the numerator and the denominator.
For the function \(f(x) = \frac{x^2 + 2}{x^2 - 16}\):

• Both the numerator and the denominator have a degree of 2, meaning they have the same degree.
• When both polynomials share the same degree, the horizontal asymptote is the ratio of the leading coefficients.
• In this function, the leading coefficient is 1 for both the numerator and the denominator, resulting in the horizontal asymptote \(y = \frac{1}{1} = 1\).

This means as \(x\) advances towards infinity or negative infinity, the value of \(f(x)\) will get closer and closer to 1.

Rational Functions

A rational function is defined as the quotient of two polynomials, represented generally as \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. The key characteristics of rational functions include their domains, vertical asymptotes, and horizontal asymptotes, which all depend on the behavior of the numerator and denominator.

• The domain of a rational function is the set of all real numbers except those where the denominator \(Q(x)\) equals zero, because division by zero is undefined.
• Vertical asymptotes occur at these undefined points when \(P(x)\) is non-zero.
• Horizontal asymptotes can be determined by comparing the degrees of \(P(x)\) and \(Q(x)\).

Understanding these aspects makes analyzing and graphing rational functions more comprehensible, helping students to predict the general behavior of the graph based on these characteristics.