Problem 23 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{3}{x^{2}-5} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = \pm\sqrt{5}\); Horizontal asymptote: \(y = 0\).

Step 1: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator of a function is equal to zero, causing the function to be undefined. To find these, solve the equation \(x^2 - 5 = 0\). This simplifies to \(x^2 = 5\). Taking the square root of both sides, we get \(x = \pm\sqrt{5}\). Thus, the vertical asymptotes are at \(x = \sqrt{5}\) and \(x = -\sqrt{5}\).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials. Here, the degree of the numerator \(3\) is 0, and the degree of the denominator \(x^2 - 5\) is 2. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). Thus, the horizontal asymptote is at \(y = 0\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in rational functions, which are fractions where the top (numerator) and bottom (denominator) are polynomials. These asymptotes are vertical lines that the graph of the function approaches but never touches or crosses.
They arise where the denominator is zero because dividing by zero makes the function undefined.

• To find the vertical asymptotes, solve the equation set by the denominator equal to zero.
• For example, consider the function given: \( f(x) = \frac{3}{x^{2} - 5} \).
• Locate the vertical asymptotes by solving \( x^{2} - 5 = 0 \).

This gives us \( x^{2} = 5 \), leading to \( x = \pm \sqrt{5} \).
Hence, the vertical asymptotes are at \( x = \sqrt{5} \) and \( x = -\sqrt{5} \). These locations on the x-axis create boundaries where the function's value heads toward infinity or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that a graph approaches as the x-values extend towards positive or negative infinity. For rational functions, they describe end behavior rather than local behavior near specific points.

• You can determine these by comparing the highest power (degree) of x in the numerator and the denominator.
• If the degree of the numerator is less than that of the denominator, the horizontal asymptote is located at \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is at the ratio of the leading coefficients.

With \( f(x) = \frac{3}{x^2 - 5} \), recognize that the numerator's degree is 0 and the denominator's degree is 2.
Therefore, the horizontal asymptote is \( y = 0 \), showing that the function levels off to 0 as x moves toward infinity.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. They can exhibit interesting features like asymptotes and holes, directly tied to their structure. Understanding these aspects can profoundly help in graphing and comprehending their overall behavior.

• The function \( f(x) = \frac{3}{x^2 - 5} \) is a rational function because it is defined as one polynomial divided by another.
• Critical traits of rational functions include their asymptotes and the potential for undefined points where their denominator equals zero.

By analyzing rational functions, one can identify the vertical asymptotes through the denominator and horizontal asymptotes based on the polynomial degrees.
This deep dive offers an intuitive look at how these functions behave, equipping you to tackle similar problems effectively.