Problem 27 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. Suppose that \(f(x)=P(x) / Q(x),\) where \(P\) and \(Q\) are polynomials with no common factors. If \(y=5\) is a horizontal asymptote for the graph of \(f,\) then \(P\) and \(Q\) have the same degree.

Step-by-Step Solution

Answer

True, because the degrees of \( P \) and \( Q \) must be equal for \( y = 5 \) to be a horizontal asymptote.

Step 1: Understanding Horizontal Asymptotes

A horizontal asymptote of a rational function \( f(x) = \frac{P(x)}{Q(x)} \) depends on the degrees of the polynomials \( P(x) \) and \( Q(x) \). If \( m \) is the degree of \( P(x) \) and \( n \) is the degree of \( Q(x) \), then:- If \( m < n \), \( y = 0 \) is a horizontal asymptote.- If \( m > n \), there is no horizontal asymptote.- If \( m = n \), the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \) respectively.

Step 2: Analyze the Given Condition

The given condition is that \( y = 5 \) is a horizontal asymptote for the graph of \( f \). This indicates that the horizontal asymptote results from \( m = n \), where the degrees of \( P(x) \) and \( Q(x) \) are equal. Additionally, the leading coefficients ratio \( \frac{a}{b} = 5 \), which aligns with the form when degrees are equal.

Step 3: Conclude Final Answer

Since for \( f(x) = \frac{P(x)}{Q(x)} \) to have a horizontal asymptote of \( y = 5 \), \( P \) and \( Q \) must indeed have the same degree \( m = n \). Hence, the statement that \( P \) and \( Q \) have the same degree given \( y = 5 \) as a horizontal asymptote is true.

Key Concepts

Rational Function

A rational function is a fraction where both the numerator and the denominator are polynomials. Consider it as a division of two polynomial expressions. For example, in the function \( f(x) = \frac{P(x)}{Q(x)} \), \(P(x)\) represents the numerator polynomial and \(Q(x)\) the denominator polynomial. It's important that \(Q(x)\) is not zero because division by zero is undefined in mathematics.Rational functions can display a variety of interesting behaviors in their graphs, such as asymptotes and intercepts. - **Horizontal asymptotes** occur based on the degree of the polynomials, which we will discuss in more detail.- **Vertical asymptotes** occur at values of \(x\) where \(Q(x) = 0\). Understanding the structure of rational functions helps to predict and comprehend how graphs behave in different situations.

Degree of Polynomial

The degree of a polynomial is a vital concept when analyzing functions, especially rational ones. The degree refers to the highest exponent of \(x\) present in the polynomial when it is expressed in its standard form.For example, for a polynomial like \(P(x) = 3x^4 + 2x^3 - x + 7\), the degree is 4 because the highest exponent is 4.In rational functions, the relation between the degrees of the numerator \(P(x)\) and the denominator \(Q(x)\) determines the graph's horizontal asymptote behavior:- **If the degree of \(P(x)\) \( (m)\) is less than the degree of \(Q(x)\) \( (n)\)**, the horizontal asymptote is \(y = 0\).- **If the degree of \(P(x)\) is greater than \(Q(x)\)**, there is no horizontal asymptote.- **If \(m = n\)**, the horizontal asymptote will be determined by the leading coefficients, resulting in \( y = \frac{a}{b} \). Understanding the degree of polynomials aids in predicting these aspects of rational functions.

Leading Coefficients

The leading coefficient is the number in front of the variable with the highest exponent in a polynomial. It plays a crucial role in determining the behavior of rational functions, particularly their horizontal asymptotes.For a polynomial \(P(x) = 4x^3 + 2x^2 + x + 5\), the leading coefficient is 4, associated with the highest degree term \(x^3\).In the context of rational functions and horizontal asymptotes:- If the degrees \(m\) and \(n\) of \(P(x)\) and \(Q(x)\) are the same, the horizontal asymptote of the graph is given by \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of \(P(x)\) and \(Q(x)\) respectively.This means that, for a rational function where \(m = n\), evaluating the horizontal asymptote involves a simple ratio of these leading coefficients, providing an immediate picture of the end behavior of the graph.