Question

True or False? In Exercises \(65-68,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of polynomial functions have no vertical asymptotes.

Short answer

True. The statement is correct. The graphs of polynomial functions have no vertical asymptotes.

Step-by-step solution

Understanding Polynomial Graphs

A polynomial function is an algebraic function consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For the provided statement, the underlying assumption is that polynomial graphs don't have vertical asymptotes.

What Are Asymptotes?

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. They come into existence when the denominator of the function goes to zero but not the numerator, thus making the function approach infinity or negative infinity.

Applying the Definitions

For polynomial functions, there is no denominator going to zero since they are not rational functions (functions that can be expressed as a quotient of two polynomials). Hence, polynomial functions don't have vertical asymptotes.

Key concepts

Vertical Asymptotes

Vertical asymptotes are important features in the study of graphs of functions. They are vertical lines on a graph that the function approaches but never actually touches or crosses. These occur particularly in rational functions, where the denominator, and not the numerator, becomes zero at some x-values, driving the function towards infinity or negative infinity.
Such lines are often expressed as equations in the form of \(x = a\), where \(a\) is the x-value causing the denominator to be zero. For example, in the function \( f(x) = \frac{1}{x} \), the denominator is zero at \(x = 0\), resulting in a vertical asymptote along the line \(x = 0\).
It's significant to note that vertical asymptotes are indicative of a function's behavior as \(x\) approaches particular values. They do not imply constraints on the function elsewhere nor do they affect other x-values across the graph. Recognizing where, and why, vertical asymptotes occur can aid in understanding function continuity and limitations.

Rational Functions

Rational functions are formed by the division of two polynomials, commonly noted as \(f(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomial expressions.
The key element of rational functions is their potential to have undefined points at the values where the denominator, \(q(x)\), becomes zero. These points often lead to vertical asymptotes, indicated by the undefined behavior or infinite tendencies at those specific x-values.
For instance, if you consider the rational function \(g(x) = \frac{x^2 + 3x + 2}{x-1}\), the denominator becomes zero at \(x=1\). Thus, this function would have a vertical asymptote at \(x = 1\).

• This differentiation of rational functions from polynomial functions is crucial since polynomial functions do not feature division by a variable, ruling out the possibility of vertical asymptotes.
• Rational functions can exhibit asymptotic behavior and discontinuities, vastly differentiating them from polynomial graphs.

Polynomial Graphs

Graphs of polynomial functions are smooth and continuous, representing a fundamental aspect of algebraic expressions without divisions by variable terms. These functions take the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\) where terms are solely based on non-negative integer powers of \(x\) added or subtracted together.
Due to their structure, polynomial functions are devoid of the discontinuities found in rational functions. The absence of any denominator variable terms inherently prevents the formation of vertical asymptotes. In simple terms, a polynomial function never compels its graph to stretch excessively or break near a particular x-value due to division by zero.
Understanding these properties explains why statements like "polynomial graphs have no vertical asymptotes" are true. Such functions may occasionally encounter undefined behavior at specific points, but it's never due to vertical divides.

• They exhibit a predictable, smooth curvature.
• Lack interruptions or undefined tendencies along their domains.

This knowledge allows one to anticipate a polynomial curve's general behavior on graphs, making them a fundamental component in algebra.