Question

1. Is it possible for a rational function to have no vertical asymptotes? If yes the how? If no then why?

2. What is the fundamental difference between a rational function and a polynomial function?

3. Can a rational function have no x - intercepts? If so then how?

4. Can a rational function have two y-intercepts? Why or why not?

5. Can the graph of a rational function cross the vertical asymptote? Why or why not?

Please TYPE out answer, just need detailed explanations to these answers please!

Answer 1

(1).

Precalculus

1. Is it possible for a rational function to have no vertical asymptotes? If yes the how? If no then why?

Ans: The rational function of the from is :

Where p(x) and q(x) both are the polynomial.

Thereis no vertical asymptote if the denominator q(x ) of rational function R(x) has no real roots or has only complex roots for all x .

And there is vertical asymptotes if the denominator q(x) has real roots.

There is no vertical asymptote if the degree of the numerator p(x) of the rational function is greater than the degree of the denominator q(x) . It is not possible. Rational functions always have vertical asymptotes.

(2) what is the fundamental difference between a rational function and a polynomial function?

Ans:

Polynomial function use only whole number as exponents of veritable.

Rational functions use variables with exponents in the denominator. But a fraction with exponents in the denominator is equivalent to using a negative exponent and this violates the definition of a polynomial.

A polynomial function is always written as a rational function.

But a rational functions may or may not be written as a polynomial function.

is a rational functions . not a polynomial function.

(3) Can a rational function have no x - intercepts? If so then how?

Ans:

The rational functions of numerator is one and will never be zeros and so this function will have no x-intercepts. Again, the graph will get very close to the x-axis but it will never touch or cross it.

Example:

and

has no x- intercepts.

(4) Can a rational function have two y-intercepts? Why or why not?

Ans:

Put the value x = 0 into the rational function and determine the value of f(x) to find the y-intercept of the rational function.

If the rational functions is square roots of numerator or denominator or both,then the rational functions has two y-intercepts.

Example:

Two y- intercepts are (0,3/2) and (0,-3/2)

(5) Can the graph of a rational function cross the vertical asymptote? Why or why not?

Ans:

The graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote.

Example:

The vertical asymptote is ,

It never cross the graph of the rational function .