True or False. Every rational function has at least one

Result:
- If there exist any values of $x$ that make the denominator of $f(x)$ equal to zero, then $f(x)$ may have vertical asymptotes at those values.
- If there are no values of $x$ that make the denominator of $f(x)$ equal to zero, then $f(x)$ does not have any vertical asymptotes.
Hence, we can conclude that the statement 'Every rational function has at least one vertical asymptote' is False.
Solution:
To solve the statement 'True or False. Every rational function has at least one vertical asymptote,' we need to consider the definition of a rational function and the conditions for the existence of vertical asymptotes.
A rational function is defined as the ratio of two polynomials, where the denominator polynomial is not equal to zero. Let's assume our rational function is denoted as $f(x)$.
To determine if a rational function has vertical asymptotes, we need to check 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 particular value. This can happen if the denominator of the rational function becomes zero for certain values of $x$.
Therefore, for a rational function $f(x)$ to have a vertical asymptote, we need to find the values of $x$ that make the denominator of $f(x)$ equal to zero. These values are the potential vertical asymptotes.

Step 1: A rational function is defined as the ratio of two polynomial functions, where the denominator is not equal to zero. To find vertical asymptotes of a rational function, we need to identify the values of $x$ for which the denominator becomes zero.
Let's consider a general rational function of the form:
$f(x)=\frac{P(x)}{Q(x)},$
where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not equal to zero.
Step 2: The vertical asymptotes occur when the denominator $Q(x)$ becomes zero. Therefore, to find the vertical asymptotes, we need to solve the equation $Q(x)=0$.
If $Q(x)$ has any real roots, then those roots correspond to vertical asymptotes. However, it is possible for a rational function to have no real roots for the denominator $Q(x)=0$. In such cases, there would be no vertical asymptotes.
Therefore, the statement Every rational function has at least one vertical asymptote is false, as there are cases where a rational function has no vertical asymptotes.

To solve the statement 'True or False. Every rational function has at least one vertical asymptote,' we can provide a counterexample to show that it is false.
Consider the rational function $f(x)=\frac{1}{x}$. This function does not have any vertical asymptotes. The denominator $x$ can be equal to 0, but as we approach $x=0$, the function tends to positive or negative infinity rather than approaching a finite value. Hence, $f(x)$ does not have a vertical asymptote.
Therefore, the statement is false, and we have provided a counterexample to prove it.