DOMAIN AND RANGE OF RATIONAL FUNCTIONS

Domain of a Rational Function

Let y  =  f(x) be a function.

Domain is all real values of x for which y is defined.

If there is any value of x for which y is undefined, we have to exclude that particular value from the set of domain. 

Example :

Let us consider the rational function given below.

y  =  1 / (x - 2)

In the above rational function, let us equate the denominator 'x  - 2' to zero.

x - 2  =  0

x  =  2

In the function

y  =  (x² - x - 2) / (x - 2),

if x  =  2, then the denominator becomes zero and the value of 'y' becomes undefined.  

So, 'y' is defined for all real values of 'x' except x  =  2.

Therefore, the domain is

R - {2}

Range of a Rational Function

Let y  =  f(x) be a function.

Range is nothing but all real values of y for the given domain (real values of x). 

Example :

Let us consider the rational function given below.

y  =  1 / (x - 2)

To find range of the rational function above, first we have to find inverse of y.

To find inverse of y, follow the steps given below.

Step 1 :

y  =  1 /(x - 2) has been defined by y in terms x. 

The same function has to be redefined by x in terms of y.

Step 2:

y  =  1 / (x - 2)

Multiply each side by (x - 2). 

(x - 2)y  =  1

xy - 2y  =  1

Add 2y to each side. 

xy  =  2y + 1

Divide each side by y.

x  =  (2y + 1) / y

Now the function has been defined by x in terms of y.

Step 3:

In x  =  (2y + 1) / y, we have to replace x by y^-1 and y by x.

Then, 

y^-1  =  (2x + 1) / x

Step 4:

Now, find the domain of y^-1.

In the inverse function y^-1, if we substitute 0 for x, the denominator becomes zero and the value of y^-1 becomes undefined.

y^-1 is defined for all real values of x except zero.

So, the domain of y^-1 is 

R - {0}

And we already know the fact that

Range (y)  =  Domain (y^-1)

Therefore, the range of y is

R - {0}

Another Way to Find Range of Rational Functions

For some rational functions, it is bit difficult to find inverse function. In that case, we have to sketch the graph of the rational function using vertical asymptote, horizontal asymptote and table of values as given below.

In this way, we can easily get the range of rational functions.

Let us see, how to find range of the rational function given below. 

y  =  1 / (x - 2)

Vertical Asymptote :

To find vertical asymptote, we have to make the denominator (x - 2) equals to zero.

When we do so,   

x - 2  =  0

x  =  2

So, the vertical asymptote is  

x  =  2

Horizontal Asymptote :

In the rational function y  =  1 / (x - 2), the highest exponent of the numerator is less than the highest exponent of the denominator. 

So there is an horizontal asymptote. 

The equation of the horizontal asymtote is

y  =  0

Table of Values :

In the given rational function y  =  1/(x-2), now we have to substitute some random values for x and find the corresponding values of y. 

We have already known that the vertical asymptote is

x  =  2 

Now, we have to take some random values for x in the following intervals.

x < 2, x > 2 but not x  =  2

(Because, x  =  2 is vertical asymptote)

Graph of 'y  =  1/(x - 2)'

When we look at the above graph, the following point is very clear.

That is, the graph (in red color) of the rational function

y  =  1 / (x - 2)

appears at every real value of y except y  =  0.

From the graph, clearly, the range of y is 

R - {0}

Apart from the stuff given in this section,  if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6