Problem 41 - College Algebra

Question

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The function \(h(x) = \frac{12 x^{3}}{3 x^{2}+1}\) does not have a horizontal asymptote.

Step 1: Identify the Degrees

The degree of the numerator is 3, as shown by the highest power of \(x\), which is \(x^3\). The degree of the denominator is 2, as seen in the highest power of \(x\), which is \(x^2\).

Step 2: Apply the Rule of Thumb

For rational functions, the behavior of the function as \(x\) approaches infinity is determined by the degrees of the numerator and denominator. When the degree of the numerator is greater than the degree of the denominator, as is the case here, the function has no horizontal asymptote.

Step 3: Confirm the Result

This can also be confirmed through long division, dividing the numerator by the denominator. In this case, as \(x\) approaches infinity, the function \(h(x)\) will also approach infinity, which means it does not have a horizontal asymptote.

Key Concepts

Rational Functions

Rational functions are a type of mathematical expression represented as fractions, where both the numerator and denominator are polynomials. These functions are typical in calculus and algebra, providing insight into various behaviors of equations, such as points of discontinuity and asymptotic behavior. Understanding rational functions is crucial when examining the behaviors of graphs at their extremities, where they tend to approach certain lines or repeat certain patterns.
Rational functions are written in the form:

• \( f(x) = \frac{P(x)}{Q(x)} \)

Here, \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) must not be zero, as division by zero is undefined. This form helps analyze different characteristics like asymptotes, intercepts, and the end behavior of the graph.When you graph rational functions, critical elements like intercepts, asymptotes (vertical, horizontal, or slant), and the function’s behavior approaching infinity or negative infinity, help define its unique path and shape.

Degree of Numerator

The degree of the numerator in a rational function is identified by the largest power of \(x\) present in the polynomial found in the numerator. This is a crucial aspect since it significantly impacts the function’s behavior as \(x\) approaches infinity.

• In our example \(h(x) = \frac{12x^3}{3x^2 + 1}\), the degree of the numerator is 3.

This indicates that the term with \(x^3\) contributes the most to the values of the polynomial when \(x\) becomes very large. This dominance is important in deciding the end behavior, as higher degree polynomials grow faster as \(x\) expands or decreases.Understanding how this degree affects the function allows students to predict many aspects of its graph, notably whether or not the function will have certain types of asymptotes.

Degree of Denominator

In rational functions, the degree of the denominator is determined by the highest power of \(x\) in the polynomial in the denominator. This degree helps define the asymptotic behavior of the function.

• In the function \(h(x) = \frac{12x^3}{3x^2 + 1}\), the degree of the denominator is 2.

This value is crucial, as it interacts with the degree of the numerator to inform about the horizontal asymptote. The degree of the denominator depicts how quickly the effects of its highest power diminish comparatively to the term in the numerator when \(x\) is very large or small. In cases where the degree of the numerator exceeds the degree of the denominator, like in \(h(x)\), the function tends towards infinity. This scenario is pivotal in determining that no horizontal asymptote exists.

End Behavior

End behavior in rational functions examines how the function behaves as \(x\) approaches positive or negative infinity. The primary components that dictate this behavior are the degrees of the numerator and denominator.
Here's what you can expect based on their relationship:

• If the degree of the numerator is greater than the degree of the denominator, like in \(h(x) = \frac{12x^3}{3x^2 + 1}\), the function doesn’t have a horizontal asymptote. Instead, \(h(x)\) trends towards infinity as \(x\) moves towards infinity.
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote will be the ratio of their leading coefficients.
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).

For \(h(x)\), the degree of the numerator being greater than the denominator signals that \(x\) has more impact in the numerator, driving the function's values towards infinity and confirming the absence of a horizontal asymptote. Recognizing these traits is invaluable for graphing these functions and predicting how they behave at their edges.