True or False. Every rational function has at least one

jack89515lg

jack89515lg

Answered question

2021-12-01

True or False. Every rational function has at least one vertical asymptote.

Answer & Explanation

James Kilian

James Kilian

Beginner2021-12-02Added 20 answers

False 
It is untrue that every rational function has at least one vertical asymptote.
For instance
f(x)=x21x+1 
although there is no vertical asymptote, is a rational function.
Result:
False

Nick Camelot

Nick Camelot

Skilled2023-05-27Added 164 answers

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.
Mr Solver

Mr Solver

Skilled2023-05-27Added 147 answers

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)=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.
Eliza Beth13

Eliza Beth13

Skilled2023-05-27Added 130 answers

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)=1x. 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.

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