**Domain and Range of Rational Functions : **

In this section, you will learn, how to find the domain and range 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 y = 1/(x-2).

In the above rational function, if we substitute 2 for x, then we get

y = 1/(2-2)

y = 1/0

y = Undefined

The denominator becomes zero and the function becomes undefined for x = 2.

So, the above function is defined for all real values of x except 2.

Therefore, the domain is

R - {2}

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 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 will become zero.

So, y^{-1} is undefined.

Hence, 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}

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)

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}

After having gone through the stuff given above, we hope that the students would have understood, how to find domain and range of rational functions.

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

HTML Comment Box is loading comments...

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 **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**