Problem 84 - Calculus

Question

In Problems \(84-86,\) explain what is wrong with the statement. If \(P(x)\) and \(Q(x)\) are polynomials, \(P(x) / Q(x)\) must be continuous for all \(x\)

Step-by-Step Solution

Answer

Rational functions are not continuous where the denominator is zero.

Step 1: Identifying Polynomial Functions

Polynomials, like any sum or difference of powers of variables, are continuous for all real numbers. They do not have any breaks or holes in their graphs, regardless of the value of the variable.

Step 2: Rational Functions

A rational function, such as \(P(x) / Q(x)\), is the quotient of two polynomials. It is continuous wherever it is defined, but it may not be continuous where the denominator \(Q(x)\) is zero since division by zero is undefined.

Step 3: Understanding Discontinuity in Rational Functions

The function \(P(x) / Q(x)\) might not be continuous at points where \(Q(x) = 0\). At these points, the function is not defined, creating potential discontinuity. This results in a break or hole in the graph of the function.

Step 4: Determining the Error in the Statement

The statement claiming \(P(x) / Q(x)\) must be continuous for all \(x\) is incorrect because it does not consider the values of \(x\) that make \(Q(x)\) zero. At these points, the function is undefined and thus discontinuous.

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, connected using addition, subtraction, and multiplication. They are denoted in the form of terms like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1},..., a_0\) are constants, and \(x\) represents the variable.

Polynomials are inherently continuous functions. This means their graphs are smooth and have no breaks, holes, or jumps for any real number value of \(x\).

Key properties of polynomials include:

• Continuity across all real numbers
• No undefined points where the function doesn't exist
• Ability to be easily manipulated and combined with other polynomials

This continuous nature of polynomials leads into understanding more complex functions like rational functions, which are built from polynomials.

Rational Functions

Rational functions are a step further from polynomials. They are defined as the quotient of two polynomials, like \(\frac{P(x)}{Q(x)}\).

Rational functions are interesting because their continuity depends on whether the denominator, \(Q(x)\), is zero. If \(Q(x)\) is not zero, the function behaves continuously like a polynomial.

Some important traits of rational functions include:

• Defined only where the denominator is not zero
• Can exhibit breaks or holes at points where \(Q(x) = 0\)
• Combination of smooth polynomial behavior with potential discontinuities

Understanding when a rational function is undefined is crucial because it leads directly into issues of discontinuity and how these functions behave at different points on their graph.

Discontinuity

Discontinuity in functions arises when there is a sudden break, hole, or jump in the graph of a function. For rational functions like \(\frac{P(x)}{Q(x)}\), discontinuities occur at points where the denominator, \(Q(x)\), equals zero.

This situation causes the function to be undefined, as division by zero is not possible.

Types of discontinuities include:

• Point Discontinuity (Removable): A hole in the graph where both numerator and denominator have a common factor that makes them zero.
• Infinite Discontinuity: Occurs when the graph has a vertical asymptote due to \(Q(x) = 0\) and \(P(x) eq 0\).

Recognizing these discontinuities helps in analyzing the full behavior of rational functions and predicting where they might behave unpredictably.

Division by Zero

Division by zero is an important concept in mathematics that has significant implications for functions, especially rational functions.

In the case of rational functions, we have \(\frac{P(x)}{Q(x)}\). The function is undefined at any point \(x\) where \(Q(x) = 0\).

Key insights about division by zero include:

• Undefined Values: At points where the denominator is zero, the function cannot yield a meaningful result, resulting in a gap in the graph.
• Impact on Continuity: Causes the graph of the function to be discontinuous at these points.
• Handling in Equations: Must be carefully considered to avoid errors in solving equations involving rational expressions.

Understanding division by zero helps explain why the original statement about \(\frac{P(x)}{Q(x)}\) being continuous for all \(x\) is incorrect, as it does not account for the discontinuities caused by zero in the denominator.