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The domain of every Rational function is the set of all Real numbers.

This question aims to find whether the domain of all the rational numbers is a set of all real numbers or not. We have to find whether this statement is true or false.

Any number that exists in the world and that can be seen falls in the category of real numbers. Real numbers include all rational, irrational, and integers except the complex numbers that are in the form of iota. Real numbers are the set of all infinite numbers that are not complex. For example: 4.0, 5, -8, 56.88 $ \sqrt 6 $ etc.  The complex numbers like $ 2 + i $, $ \sqrt {6 } i – 9 $

Real numbers are often written as R = $ Q \cup Q’ $ which means the set of all the rational numbers union the set of all irrational numbers is called real numbers.

There are generally two types of real numbers as all numbers are either rational or irrational.

Rational numbers:

Any number represented as the quotient of numerator and denominator is called a rational number. Rational numbers often take the form of $ \frac { p } { q } $. The p in the quotient is the numerator while the q is the denominator which is always a non-zero value. The numerator can be in the form of any integer, natural number, whole number, or decimal. For example, 3.9, 0.8, 1.666, $ \frac { 2 } { 7 } $, $ \ frac { -8 } { 9 } $ etc

Expert Answer

Every Rational number is a real number but the domain of the rational numbers is not always the set of all real numbers. The domain of the rational numbers is the set of all real numbers where the function is defined. If zero is included in the denominator then it is not the domain.

For example, if we take a function  $ f ( x) $ and its domain is $ g ( \frac { 1 } { x } ) $ then it can be written as:

\[  f  (  x  )  =  \frac  {  1  }  {  x  }  \]

If we put values of x in the function:

\[ f  (  4  ) = \frac  {  1  }  {  4 } \]

\[ f  (  3  ) = \frac  {  1  }  {  3 } \]

\[ f  (  5  ) = \frac  {  1  }  {  5 } \]

Then the domains of the functions are $ \frac  {  1  }  {  4 } $, $ \frac  {  1  }  {  3 } $ , $ \frac  {  1  }  {  5 } $ and above-mentioned statement becomes false.

Numerical Results

The domain of all the rational numbers is a set of all real numbers that is not true; no vertical asymptote and hole is formed on the graph. 

Example

If we put the following expressions in the function:

\[  f  (  x  )  =  \frac  {  1  }  {  x  }  \]

\[  f  (  1  +  3  x  )  =  \frac  {  1  }  {  1  +  3  x  }  \]

The domain of all the rational numbers is a set of all real numbers that is not true as no vertical asymptote and hole is formed on the graph. 

Image/Mathematical drawings are created in Geogebra.

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