Problem 7 - Calculus

Question

In Exercises \(3-10\), determine the end behavior of each function as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). $$f(x)=\frac{3 x^{2}+5 x+6}{x^{2}-4}$$

Step-by-Step Solution

Answer

As \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\), \(f(x)\) approaches 3.

Step 1: Identify the Leading Terms

The leading terms in the numerator and the denominator of the given function are the highest degree terms. In this case, the leading term of the numerator is \(3x^2\) and for the denominator is \(x^2\). These terms will dictate the end behavior of the rational function.

Step 2: Compare the Degrees of the Polynomials

Both the numerator and the denominator have the degree of 2, which is the same. When the degrees of the numerator and the denominator are equal, the end behavior of the function is determined by the ratio of the leading coefficients of these terms.

Step 3: Determine the End Behavior Ratio

The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the ratio of these coefficients is \(\frac{3}{1} = 3\).

Step 4: Conclude End Behavior for \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\)

Since the degrees of the numerator and the denominator are equal, the function \(f(x)\) approaches the horizontal asymptote \(y = 3\) as \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\). This means for very large positive or negative values of \(x\), \(f(x)\) behaves like 3.

Key Concepts

Rational functions

A rational function is a mathematical expression represented as the fraction of two polynomials. In simple terms, it is a division of one polynomial by another. This form can be expressed as \[ f(x) = \frac{p(x)}{q(x)} \]where:

• \(p(x)\) is the numerator and is a polynomial.
• \(q(x)\) is the denominator and is also a polynomial.

An important aspect of rational functions is that the denominator cannot be zero, as division by zero is undefined in mathematics.
When studying rational functions, you'll often encounter various features like vertical asymptotes, holes, and horizontal asymptotes. These elements are essential for understanding the behavior of the function. Unlike polynomial functions, rational functions may not be defined for all values of \(x\) because the denominator can make the function undefined if it equals zero. Understanding the general behavior of rational functions helps in analyzing their graphs effectively.

Horizontal asymptotes

Horizontal asymptotes play a crucial role in determining the end behavior of rational functions. They indicate the value that a function approaches as \(x\) tends to positive or negative infinity. Basically, they're like invisible lines that the function "approaches" but never really touches, at least at infinity.

• When the degree of the polynomial in the numerator is less than the denominator, the horizontal asymptote is \(y = 0\).
• When the degrees are the same in both the numerator and the denominator, as it was in our example with the function \(f(x) = \frac{3x^2 + 5x + 6}{x^2 - 4}\), the horizontal asymptote is determined by the ratio of the leading coefficients. For this case, the horizontal asymptote is \(y = \frac{3}{1} = 3\).
• If the polynomial in the numerator's degree is greater than the denominator, there's no horizontal asymptote, but there might be an oblique (slant) asymptote instead.

In practical terms, if you're graphing a rational function, identifying the horizontal asymptotes helps you see what happens to the graph as \(x\) moves to infinity or negative infinity.
Remember, horizontal asymptotes describe end behavior, they don't impact the function's behavior at specific finite points.

Polynomial degrees

Polynomial degrees are an important concept when working with rational functions since they help determine the function's behavior at infinity. The degree of a polynomial is the highest power of the variable in the polynomial equation.

• For example, in the polynomial \(3x^2 + 5x + 6\), the degree is 2 because the highest power of \(x\) is 2.
• Similarly, for \(x^2 - 4\), the degree is also 2.

Degrees help us predict how the function behaves when \(x\) becomes very large or very small. They are crucial when looking for horizontal asymptotes in rational functions.
By comparing the degrees of the numerator and the denominator:

• If both degrees are equal, the horizontal asymptote is the ratio of their leading coefficients.
• If the numerator's degree is less, the horizontal asymptote is \(y = 0\).
• If the numerator's degree is greater, there are no horizontal asymptotes (but possibly oblique asymptotes).

Understanding polynomial degrees simplifies the process of evaluating rational functions and provides a clear picture of the end behavior of the function. You can analyze the trend without making numerous computations, saving time and enhancing clarity in graphing and solving.