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.

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