Problem 29 - College Algebra

Question

In Exercises \(29-36,\) find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{12 x}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x)=\frac{12x}{3x^{2}+1}\) is the line \(y = 0\).

Step 1: Identify the degree of the polynomial in the numerator and the denominator

First, observe the function \(f(x)=\frac{12x}{3x^{2}+1}\). The degree of the polynomial 12x (the numerator) is 1. The degree of the polynomial \(3x^{2}+1\) (the denominator) is 2.

Step 2: Compare the degrees

In this step, compare the degrees of the polynomials in the numerator and the denominator. If the degree of the denominator is strictly greater than the degree of the numerator, as in this case, then the x-axis (the line y = 0) is the horizontal asymptote.

Step 3: Conclusion

For \(f(x)=\frac{12x}{3x^{2}+1}\), the degree of the denominator \(3x^{2}+1\) is 2, which is greater than the degree of the numerator 12x, which is 1. So, the x-axis or the horizontal line y = 0 is the horizontal asymptote for this function.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that represent the division of two polynomials. They take the form of \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \eq 0 \). The function \( f(x)=\frac{12x}{3x^{2}+1} \) in our exercise is a typical example of a rational function, consisting of a linear polynomial in the numerator and a quadratic polynomial in the denominator.

Rational functions often feature interesting behaviors as the variable approaches certain values or infinity. Some of these behaviors include vertical and horizontal asymptotes, which are boundary lines that the graph of the function approaches but does not cross. In the case of \( f(x)=\frac{12x}{3x^{2}+1} \), understanding its horizontal asymptote is crucial to predicting its long-term behavior as \( x \) becomes very large or very small.

Polynomial Degree Comparison

When analyzing rational functions for asymptotic behavior, a vital step is comparing the degrees of the numerator and denominator polynomials. The 'degree' of a polynomial is the highest power of \( x \) in its terms. For instance, in \( f(x) = \frac{12x}{3x^{2}+1} \), the numerator \( 12x \) has a degree of 1, while the denominator \( 3x^{2}+1 \) has a degree of 2.

This degree comparison can determine the horizontal asymptote of the function. If the degree of the numerator is less than the degree of the denominator, the x-axis (\( y = 0 \) line) becomes the horizontal asymptote. This approach simplifies identifying the end behavior of the function without extensively plotting its graph or calculating limits.

Asymptotic Behavior

The concept of asymptotic behavior refers to the way a function behaves as it moves towards infinity or a particular point. For rational functions, horizontal asymptotes are particularly interesting as they indicate the value that the function approaches but never actually reaches as \( x \) tends towards infinity or negative infinity.

Our function, \( f(x) = \frac{12x}{3x^{2}+1} \), illustrates that when the degree of the denominator is higher than that of the numerator, the function approaches the x-axis as \( x \) becomes very large in either the positive or negative direction. Consequently, the horizontal asymptote for this function is the line \( y = 0 \). It's crucial to recognize that the function doesn't flatten out at \( y = 0 \) but gets infinitely close, defining a boundary of sorts for its long-term behavior. Understanding this concept helps in graphing rational functions and in predicting the function's behavior without detailed calculations.