Emily-Jane Bray

2021-08-10

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.

Nicole Conner

The given statement is "graphs of polynomial functions have no vertical asymptotes".
It is known that every polynomial of degree n is continuous on the set of all real number,
hence the graph of any polynomial does not have any holes and hence they does not have any asymptote. NKS Note that the asymptote x=c exists only when the function approaches$±\mathrm{\infty }$while x approaches c or the function approaches $-\mathrm{\infty }$
while x approaches ${c}^{+}\left(\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{c}^{-}\right)$ and approaches oo while x approaches ${c}^{+}\left(\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{c}^{-}\right)$ But for any real c,
$\underset{x\to {c}^{+}}{lim}{P}_{n}=\underset{x\to {c}^{-}}{lim}{P}_{n}={P}_{n}\left(c\right)\ne \mathrm{\infty }$
Thus, the given statement is true.

star233

The statement is false. Polynomial functions can have vertical asymptotes under certain conditions.
A polynomial function is defined as a function of the form $f\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\dots +{a}_{1}x+{a}_{0}$, where ${a}_{n},{a}_{n-1},\dots ,{a}_{1},{a}_{0}$ are coefficients and $n$ is a non-negative integer.
If the degree of the polynomial is even (i.e., $n$ is an even number), then the graph of the polynomial function will not have any vertical asymptotes. This is because the graph of an even-degree polynomial is symmetric about the y-axis, and it approaches infinity or negative infinity as $x$ approaches positive or negative infinity, respectively.
However, if the degree of the polynomial is odd (i.e., $n$ is an odd number), then the graph of the polynomial function can have vertical asymptotes. An example of such a polynomial is $f\left(x\right)={x}^{3}$, whose graph has a vertical asymptote at $x=0$.
Therefore, the statement ''The graphs of polynomial functions have no vertical asymptotes'' is false, as polynomial functions of odd degree can have vertical asymptotes.

user_27qwe

The statement is $\text{true}$.
Explanation: Polynomial functions do not have vertical asymptotes. A vertical asymptote occurs when the graph of a function approaches a vertical line as the input values approach a certain value. However, polynomial functions are continuous and smooth, and their graphs do not exhibit vertical asymptotes.

karton

Result:
Polynomial functions can have vertical asymptotes under specific conditions, but in general, they do not have them.
Solution:
$f\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\dots +{a}_{1}x+{a}_{0}$ where ${a}_{n},{a}_{n-1},\dots ,{a}_{1},{a}_{0}$ are coefficients, and $n$ is a non-negative integer.
To find vertical asymptotes, we need to examine the behavior of the function as $x$ approaches certain values. A vertical asymptote occurs when the function approaches positive or negative infinity as $x$ approaches a specific value.
For polynomial functions, it is possible to have vertical asymptotes under certain conditions. Specifically, a polynomial function can have a vertical asymptote if there exists a vertical line $x=c$ such that $f\left(x\right)$ approaches positive or negative infinity as $x$ approaches $c$.
However, in general, polynomial functions do not have vertical asymptotes. The reason is that polynomial functions are continuous and smooth curves that can be defined for all real values of $x$. They do not have abrupt changes or vertical gaps that would lead to vertical asymptotes.
Hence, the statement is false. Polynomial functions can have no vertical asymptotes, but they are not restricted from having them entirely.
For example, let's consider the polynomial function $f\left(x\right)={x}^{2}$. This function does not have any vertical asymptotes since it is defined for all real values of $x$. As $x$ approaches positive or negative infinity, $f\left(x\right)$ also approaches positive infinity, but there is no specific value of $x$ where $f\left(x\right)$ approaches infinity.

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