What is a rational function? What are Asymptotes?

Question

Give examples of rational functions, with vertical asymptotes, horizontal asymptote.

State Domain or range of the function selected as an

example, show and explain how domain and range can be written in inequality notation, interval

notation, on a number line. Explain how would you graph a rational function?. Model a function

with Vertical Asymptote at x=3, and hole at x=5, horizontal asymptote at y= 4.

Answer 1

Step 1 :

 

Rational function:  A rational function is ratio of two polynomial functions.

A rational function is in form of , where and are polynomial function.

Asymptote: A straight line that continually approaches a given curve but does not meet it at any finite distance.

There are three types of asymptotes.

1) Vertical asymptote.

2) Horizontal asymptote.

3) Oblique/ Slant asymptote.

Example of a rational function with vertical and horizontal asymptote is .

Observe the rational function :

Vertical asymptote of the rational function are obtained by equating denominator to zero.

Vertical asymptote is .

If the degree of the numerator and the denominator are equal, then horizontal asymptote is defined as

.

Therefore horizontal asymptote is .

Step 2 :

The rational function is .

Domain :

Domain is set of values of x which makes the function mathematically correct.

Denominator of the function should not zero.

Domain of the function is all values of x except .

Domain on number line :

Note : The hallow circle indicates that the value is not included in the domain.

Domain in inequality notation : .

Domain in interval notation : .

Range :

Range is output value of function.

Range of the function is all values of x except .

Range on number line :

Note : The hallow circle indicates that the value is not included in the range.

Range in inequality notation : .

Range in interval notation : .

Solution :

Example of a rational function is .

Vertical asymptote is .

Horizontal asymptote is .

Domain in inequality notation : .

Domain in interval notation : .

Range in inequality notation : .

Range in interval notation : .

Answer 2

(b)

Step 1:

Graphing of a rational function :

1) Determine the domain by setting the denominator equal to zero.

2) Determine vertical asymptote(s). Since the rational function is already in simplest form, the vertical asymptote(s) will occur at the domain restriction(s).

3) Determine the horizontal asymptote.

4) Determine the x and y-intercepts.

5) Plot other points.

6) Graph the function following from the step 1 to step 5.

Step 2:

Example of a rational function with vertical and horizontal asymptote is .

Domain of the function : .

Vertical asymptote is .

Horizontal asymptote is .

Determine the x and y-intercepts.

Find the x-intercept of the function by substituting y=0.

x-intercept is .

Find the y-intercept of the function by substituting x=0.

y-intercept is .

Step 3:

Draw the table for different values of x.

Graph :

Graph the rational function using following specifications :

Domain of the function : .

Vertical asymptote is .

Horizontal asymptote is .

x-intercept is .

y-intercept is .

Plot the points obtained in the table.

Solution :

Graph of the rational function is .

.

Answer 3

(c)

Step 1:

 

The function has vertical Asymptote at x =3 and hole at x =5.

Hence the denominator of the function is .

Horizontal asymptote is at y = 4.

If the degree of the numerator and the denominator are equal, then horizontal asymptote is defined as

.

Hence the numerator of the function is assumed as .

Therefore, sample answer is .

Solution :

Sample answer is .