## Problem 671

For the pair of function f and g, determine the domain of f/g.

F(x) = ,g(x) = 4-x

Solution:-

The domain of t/g is the set of all values common to the domains of t and g, excluding values for which g(x) is 0.

The domain of a function defined by an equation is the set of all number for which real values of the function can be calculated. A function can be undefined at a number because calculating its value results in an undefined in an undefined operation like division by zero or an even root of a negative number.

For the function , f(x) =. The denominator is zero for x= 3, which result in an undefined value. Therefore, the domain of f is {x I x is  a real number and x3}.

For the function , g(x) = 4 – x, a value can be calculated for any real number x. Therefore, the domain of g is {x I x is a real number}.

The domain of the quotient also excludes all values for which f(x) is zero. Therefore solve the equation

4 – x = 0.

X = 4

The domain of f/g is the set of all values common to { x I x is a real number and x 3} and {x I x is a real number}, and also excluding x = 4.

Therefore , the domain of f/g is {x I x is a real number and x 3 and  x 4}.

## Find domain and range

Find domain and range

For every relation, there are two important sets of elements called the domain and range.

Domain and Range

In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable ( y) is the range.

Example

{(3, 21), (4, 2), (4, 5), (6, 8)}

The domain, the set of x-values, is {3, 4, 6}; the range, the set of y-values, is {21, 2, 5, 8}. This relation is not a function because the same x-value 4 is paired with two different y-values, 2 and  5.

## What is relation and types of relations

What is relation (Mathematics) and types of relations:-

Relation (mathematics) is same as in real life, it is define as two things connected to each other or it is the property of things to connect in a certain manner.

Example:

• Akbar was the son of Humayun
• 25 is the multiple of 5.
• In a triangle ABC, AB is the base.

Domain and Range of a Relation:  If R is a relation defined between set A and set B then set of first elements of ordered pair is known as Domain and set of second elements of the ordered pair is known as Range means Domain of R is {a|(a,b) є R} and Range of R is {b|(a,b) є R}.

Example :

If R = { (1,2),(1,4),(1,6),(2,2),(2,4),(2,6)} ( A relation between A and B )

So Domain of R  = {1, 2} = A  and Range of R = {2,4,6} = B

Types of Relation:-

1. Reflexive Relation:  If every element of a set A is related to itself In a Relation R it means that Relations R is reflexive relation.

(a,a) є R for all a є A.

2. Symmetric Relation: A relation R defined in a set A, if element ‘a’ is related to ‘b’ and ‘b’ is also related to ‘a’ in the same way then this relation is known as Symmetric Relation.

Relation R is symmetric if (a,b) є R => (b,a) є R for all a,b є A.

3. Anti symmetric Relation: A relation R defined in a set A. if element ‘a’ is related to ‘b’ and ‘b’ is related to ‘a’, both will be true only when a=b, then this relation is known as anti symmetric Relation.

(a,b) є R and (b,a) є R => a = b, for all a,b, є A

4. Transitive Relation: A relation R defined in a set A, if element ‘a’ is related to ‘b’ and ‘b’ is related to element ‘c’ from this if ‘a’ has relation to c then this relation is known as Transitive relation.

If (a,b)є R and (b,c)є R => (a,c)єR , for all a,b,c, є A

5. Equivalence Relation: we call equivalence relation  to a Relation R if

R is reflexive

R is Symmetric

R is Transitive

6. Partial order Relation:  we call Partial order relation  to a Relation R if

R is reflexive

R is Anti Symmetric

R is Transitive

7. Total order Relation: we call Partial order relation  to a Relation R if

R is Partial Order relation

And for every a,b є A either (a,b) є R or (b,a) є R or a = b are true.