Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. A rational function can have infinitely many vertical asymptotes.

Step-by-Step Solution

Answer

The statement is false. A rational function can have only a finite number of vertical asymptotes.

Step 1: Define the rational function

A rational function is any function which can be defined by a rational fraction, i.e., an algebraic fraction such that both the numerator and the denominator are polynomials. Typically, this form looks like \(f(x) = \frac{P(x)}{Q(x)}\), where \(P\) and \(Q\) are polynomials.

Step 2: Understand asymptotes

Vertical asymptotes of a rational function are vertical lines, typically written in the form \(x=c\), where \(c\) is any real number. For a rational function, vertical asymptotes occur at the \(x\) values which make the denominator of the function equal to zero (given that these \(x\) values don't also make the numerator equal to zero). This is because when the denominator of a fraction is zero, the fraction (function) is undefined.

Step 3: Analyze the statement

The question is whether a rational function can have infinitely many vertical asymptotes. A rational function has vertical asymptotes where the denominator polynomial is zero, provided the same values do not make the numerator zero. As the numerator and denominator are polynomials and a polynomial of degree \(n\) can have at most \(n\) real roots, the denominator can be zero at only a finite number of points. Thus, a rational function can have only a finite number of vertical asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are significant in the behavior of rational functions. Imagine a vertical line that a function approaches as it goes to positive or negative infinity, but never actually crosses. This line is known as a vertical asymptote. In mathematical terms, if you have a rational function \( f(x) = \frac{P(x)}{Q(x)} \), where \( P \) and \( Q \) are polynomials, then the vertical asymptotes occur where the polynomial in the denominator, \( Q(x) \), equals zero. At these points, the function becomes undefined.

• If \( Q(x) = 0 \), then \( x \) can be a vertical asymptote.
• Vertical lines, like \( x = c \), represent these asymptotes.

It is vital to note, an asymptote occurs only if the corresponding \( x \) value doesn’t make the numerator zero too. If \( P(x) = 0 \) as well, then you have a removable discontinuity instead of a vertical asymptote.

Polynomials

Polynomials are essential in defining rational functions because they form the backbone of the numerator and denominator. A polynomial is a mathematical expression composed of variables (often \( x \)) raised to non-negative integer powers and multiplied by coefficients. They can simply be constants or have many terms like \( 3x^2 + 2x + 1 \).

• Polynomials can have varying degrees, which is the highest power of the variable.
• The degree tells us the number of roots or solutions the polynomial can have.

When discussing rational functions, understanding the degree is crucial. The degrees of the polynomials in the numerator and the denominator influence the nature of the function, such as the number of vertical asymptotes possible.

Numerator and Denominator

In rational functions, the numerator and denominator are critical elements. Consider them as parts of a fraction representing a function, such as \( f(x) = \frac{P(x)}{Q(x)} \). Here, \( P(x) \) is the numerator and \( Q(x) \) is the denominator. They both need to be polynomials.

• The numerator determines the function values when \( Q(x) eq 0 \).
• The denominator dictates where the function might become undefined, leading to vertical asymptotes.

For a vertical asymptote to exist, the denominator must be zero and not be canceled out by the numerator zeros. Essentially, they decide the function’s domain and where any asymptotic behaviors might occur.

Undefined Functions

An undefined function comes into play when part of the function, typically the denominator, equals zero. At these points, the result of the function doesn't exist in the real number system. Particularly in rational functions, undefined points are significant as they suggest potential vertical asymptotes.

• When \( Q(x) = 0 \), the function is undefined at those \( x \) values.
• An undefined point does not guarantee a vertical asymptote if the numerator, \( P(x) \), is also zero at that point.

In practical terms, undefined functions help us identify key 'trouble spots' in the graph of the function. At these undefined points, a graph either shoots up or down, creating breaks or discontinuities.