Problem 4 - College Algebra

Question

Can a graph of a rational function have no vertical asymptote? If so, how?

Step-by-Step Solution

Answer

Yes, a rational function can lack vertical asymptotes if its denominator is a non-zero constant.

Step 1: Understanding Rational Functions

A rational function is given by the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Vertical asymptotes occur where the denominator \( Q(x) \) is equal to zero, as long as the numerator \( P(x) \) is not zero at the same point.

Step 2: Condition for No Vertical Asymptote

For a graph of a rational function to have no vertical asymptote, there must be no value of \( x \) for which \( Q(x) = 0 \). This can occur if \( Q(x) \) is a non-zero constant, meaning that \( Q(x) \) does not depend on \( x \).

Step 3: Example of No Vertical Asymptote

Consider the function \( f(x) = \frac{3x + 2}{5} \). Here, \( Q(x) = 5 \), a constant polynomial. Since \( Q(x) \) is never zero, the function does not have any vertical asymptotes.

Key Concepts

Vertical Asymptotes

When studying rational functions, it is crucial to understand the concept of vertical asymptotes. A vertical asymptote occurs in a rational function at a value of \( x \) where the function goes to infinity, either positively or negatively. This happens because the denominator of the function, \( Q(x) \), is zero, while the numerator, \( P(x) \), is not zero at the same point.

To illustrate, consider the rational function \( f(x) = \frac{P(x)}{Q(x)} \). If \( Q(x) = 0 \) at \( x = a \), and \( P(a) eq 0 \), the graph of \( f(x) \) will have a vertical asymptote at \( x = a \).

Understanding the behaviour around vertical asymptotes is important. As \( x \) approaches \( a \) from either side, the function values tend to \( \pm \infty \). This can be observed graphically, where the curve approaches a vertical line, but never touches it.

• If approaching from the left, the curve might shoot upwards or dive downwards.
• The same applies from the right side.

Vertical asymptotes signal that a portion of the graph is undefined at specific \( x \) values and therefore, help us understand the limits of rational functions.

Polynomials

Polynomials form the backbone of rational functions. They are expressions composed of variables and coefficients, constructed using addition, subtraction, multiplication and non-negative integer exponents. For example, a general polynomial looks like this: \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \).

Polynomials define both the numerator and denominator in a rational function \( f(x) = \frac{P(x)}{Q(x)} \). Understanding their structure helps us deduce conditions for vertical asymptotes.

• If the degrees of \( P(x) \) and \( Q(x) \) differ, it impacts the characteristics of the rational function's graph.
• Critical points where \( Q(x) = 0 \) dictate possible locations for vertical asymptotes.
• Roots of the polynomial play a key role in understanding intersections and symmetry.

By simplifying polynomials and evaluating where they become zero, we gain insights into rational functions' behaviour and possible vertical asymptotes.

Constant Polynomials

A constant polynomial is a special case in polynomial expressions. It consists of a single term that does not depend on \( x \). For example, \( Q(x) = c \), where \( c \) is a constant value that doesn't change.

In the context of rational functions, a constant polynomial in the denominator \( Q(x) \) means it never equals zero. Consequently, the function \( f(x) = \frac{P(x)}{Q(x)} \) has no vertical asymptotes, as seen in the example \( f(x) = \frac{3x + 2}{5} \). Since \( 5 \) is always non-zero, the function is defined for all \( x \) values.

• With constant polynomials, there are no critical points where the function is undefined.
• The graph is smooth along the entire \( x \)-axis, without any jumps or breaks.
• These functions demonstrate that rational expressions can indeed exist without vertical asymptotes.

Constant polynomials provide a reliable way to achieve stable, continuous rational functions without worrying about potential infinity jumps.