# DOMAIN AND RANGE OF RATIONAL FUNCTIONS WITH HOLES

## About the topic "Domain and range of rational functions with holes"

"Domain and range of rational functions with holes" is a much needed stuff required by almost all the students who study math in high schools.

Even though students can get this stuff on internet, they do not understand exactly what has been explained.

To make the students to understand the stuff "Domain and range of rational functions with holes", we have given step by step explanation.

## Domain of a rational function with hole

Let f(x) = (x² - x - 2) / (x-2)

Domain is nothing but the real values of "x" for which "f(x)" is defined.

In the above rational function, if we make the denominator  x -2 equal to zero, we get x = 2

That is, x - 2 = 0 ===>  x = 2

Hence, "y" is defined for all real values of "x" except x = 2

Hence, Domain (y)  = R - {2}

## Finding hole of a rational function

For the rational function f(x) = (x² - x - 2) / (x-2), let us try to find hole, if any.

To find hole, let us try to simplify the given rational function as given below. In the above simplification, the common factor for numerator and denominator is (x-2). So there is a hole.

(Note : If there  is no common factor for numerator and denominator, there is no hole)

Now we have to make the common factor (x-2)  equal to zero.

When we do so, we get

x - 2 = 0 ===>  x  =  2

So, the hole is at  x = 2

After having crossed out the common factor (x-2), the function is simplified to f(x) = x+1   or   y = x+1.

Now, if we plug x = 2 in y = x+1, we get y = 3.

Hence, the hole appears on the graph at (2 , 3)

After simplification, the given rational function becomes y = x + 1 which is linear and its graph will be a straight line.

## Graph of y=x+1 ## Range of a rational function with hole

When we look at the above figure, the graph of y= x+1 appears at every real value of "y" except at y = 3. Because, there is a hole at        y = 3.

So, the range is all real values except "3".

More clearly, Range (y) = R - {3}

You can also visit the following sites to know more about domain and range of rational functions.

http://hotmath.com

http://www.analyzemath.com

https://cims.nyu.edu

http://www.montereyinstitute.org

http://www.purplemath.com

https://socratic.org

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

WORD PROBLEMS

Word problems on simple equations

Word problems on linear 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 