Problem 30 - College Algebra

Question

Find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{15 x}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the given function is y = 0.

Step 1: Identify the Degrees

The degree of a polynomial is the highest power of x in its expression. For \(f(x)=\frac{15x}{3x^{2}+1}\), the degree of the numerator (15x) is 1 and the degree of the denominator (\(3x^{2}+1\)) is 2.

Step 2: Compare the Degrees

It's seen that the degree of the denominator is greater than the degree of the numerator. When this is the case for a rational function, the horizontal asymptote is the x-axis, which is the line y = 0.

Step 3: Write the horizontal asymptote

Based on the previous analysis, the horizontal asymptote of the function \(f(x)=\frac{15x}{3x^{2}+1}\) is y = 0.

Key Concepts

Rational Functions

Rational functions are the division of two polynomials. Just like fractions, where we divide numbers, a rational function divides one polynomial by another. They are written in the form of \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are both polynomials, and \(q(x)\) is not equal to zero.

Understanding rational functions is crucial because they describe many real-world situations, such as the rate of change of costs or the concentration of a chemical solution over time. When working with rational functions, always remember that the properties of the function, like whether it has horizontal asymptotes, depend on the relationship between the degrees of the polynomials in the numerator and the denominator.

Polynomial Degrees

The degree of a polynomial signifies the highest power of the variable that appears in the polynomial. For example, in the polynomial \(7x^3 + 5x^2 - 2x + 1\), the degree is 3 because the highest power of \(x\) is 3.

The degree plays a pivotal role when analyzing rational functions, as it helps determine their end behavior and whether they have horizontal asymptotes. If the degree of the numerator is less than the degree of the denominator, as in the exercise, the rational function will always have a horizontal asymptote at \(y = 0\). Knowing this helps us predict the behavior of the function as \(x\) approaches very large or very small values.

Graph Analysis

Graph analysis involves studying the visual representation of functions to understand their characteristics. For rational functions, this includes identifying asymptotes, intercepts, and regions where the graph changes behavior.

The graph of a rational function can reveal a lot about its properties. For instance, where the graph gets closer to but never touches a line, you've found an asymptote. It's like a boundary that the function values can approach indefinitely but never reach. In our example, the x-axis serves as a horizontal boundary for the graph as the values of \(x\) get very large in magnitude.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it heads towards infinity or negative infinity. When a function's value gets closer and closer to a specific value (in our case, the x-axis or \(y = 0\)) as \(x\) increases or decreases without bound, that specific value is called a horizontal asymptote.

This concept is critical when exploring rational functions because it informs us about the function's long-term behavior. Understanding asymptotic behavior helps mathematicians and scientists make predictions and understand limits within their respective fields. As \(x\) becomes very large (either positively or negatively), the value of our function \(f(x)\) approaches the horizontal asymptote at \(y = 0\), telling us that the graph levels off at this line.