Solved examples on Domain-I
Find the domain of the following :
Students are requested to read all previous posts before reading this.
Here is the quick look
Domain: The Possible point of for which we can get finite value of
To know more about domain and its properties (click here)
1. If then, , i.e.
2. If then,
3. If then and
4. If then and g(x) must be defined.
5. If or then
if a function is defined in some interval or points( by definition domain), and another function is defined in some other interval and points (by definition domain) .... then sum , subtraction , multiplication and division of the functions are defined on the interval or points which are common to both.
1) If domain of and are and respectively then domain of
a) ,
b) ,
c) , Since here, , therefore , i.e. we have to eliminate those points from for which , because
1.
We cannot find the root of negative numbers.
therefore,
1.
2. log is defined when,( i.e. domain of log function = )
( since log is defined only for positive values )
Take 1st,
or,
To know how to solve logarithmic inequality (click here)
since base of log is 1/2 which is less than 1,
therefore, inequality gets reversed
To know how to solve quadratic inequalities (click here)
or,
or,
or,
or,
To know how to solve this (click here)
By wavy-curvy method
..................................... (1)
2.
or,
or,
or,
by wavy-curvy method,
...................... (2)
x must satisfied both the conditions,
therefore,
..................................................................................................... (Number Line)
-1 0 1 2 3 4 5 6
<-------------------------------------------->
*********************> <*****************
clearly,
Therefore,
2.
therefore, x - [x] > 0
=> x > [x]
x is always greater than [x] except x belongs to Integers.
see the definition of greatest integer function ( click here )
when, x = Integers, x = [x]
Also, see from the above graph line y = x touches the graph y = [x] on integers i.e. on integers both the functions are equal.
Therefore,
Domain = R - I
OR you can write
therefore, {x} > 0
See from the above graph,
{x} is always greater than '0' except on integers,
because, {x} = 0, when x belongs to Integers
Therefore,
Domain = R - I
3.
Note: f(x) = g(x) + h(x) + k(x)............
then, .......................
To know more about domain and its properties (click here)
Here are three functions
1.
It is defined for all value of 'x'
Domain,
2.
since, is defined in the interval [-1,1] i.e. domain of sin inverse x is [-1,1]
therefore,
( see from the above graph that when )
or,
since, tan x is an increasing function
therefore,
or,
or,
3.
since, is defined in the interval [-1,1] i.e. domain of sin inverse x is [-1,1]
Therefore, Domain of = [-1,1]
or,
also, see the graph
Therefore, domain of ,
pi/4 = 3.14/4 < 1
and, -pi/4 > -1
therefore,
4.
To know the properties of logarithmic function (click here)
Therefore,
1. for to be defined
x > 0
or,
2. for to be defined
see the above graph,
at x = 1,
therefore, or
therefore, take the intersection of (1) and (2)
or,
Students are requested to read all previous posts before reading this.
Here is the quick look
Domain: The Possible point of for which we can get finite value of
To know more about domain and its properties (click here)
1. If then, , i.e.
2. If then,
3. If then and
4. If then and g(x) must be defined.
5. If or then
if a function is defined in some interval or points( by definition domain), and another function is defined in some other interval and points (by definition domain) .... then sum , subtraction , multiplication and division of the functions are defined on the interval or points which are common to both.
1) If domain of and are and respectively then domain of
a) ,
b) ,
c) , Since here, , therefore , i.e. we have to eliminate those points from for which , because
1.
We cannot find the root of negative numbers.
therefore,
1.
2. log is defined when,( i.e. domain of log function = )
( since log is defined only for positive values )
Take 1st,
or,
To know how to solve logarithmic inequality (click here)
since base of log is 1/2 which is less than 1,
therefore, inequality gets reversed
To know how to solve quadratic inequalities (click here)
or,
or,
or,
or,
To know how to solve this (click here)
By wavy-curvy method
..................................... (1)
2.
or,
or,
or,
by wavy-curvy method,
...................... (2)
x must satisfied both the conditions,
therefore,
..................................................................................................... (Number Line)
-1 0 1 2 3 4 5 6
<-------------------------------------------->
*********************> <*****************
clearly,
Therefore,
2.
therefore, x - [x] > 0
=> x > [x]
x is always greater than [x] except x belongs to Integers.
see the definition of greatest integer function ( click here )
when, x = Integers, x = [x]
Also, see from the above graph line y = x touches the graph y = [x] on integers i.e. on integers both the functions are equal.
Therefore,
Domain = R - I
OR you can write
therefore, {x} > 0
See from the above graph,
{x} is always greater than '0' except on integers,
because, {x} = 0, when x belongs to Integers
Therefore,
Domain = R - I
3.
Note: f(x) = g(x) + h(x) + k(x)............
then, .......................
To know more about domain and its properties (click here)
Here are three functions
1.
It is defined for all value of 'x'
Domain,
2.
since, is defined in the interval [-1,1] i.e. domain of sin inverse x is [-1,1]
therefore,
( see from the above graph that when )
or,
since, tan x is an increasing function
therefore,
or,
or,
3.
since, is defined in the interval [-1,1] i.e. domain of sin inverse x is [-1,1]
Therefore, Domain of = [-1,1]
or,
also, see the graph
Therefore, domain of ,
pi/4 = 3.14/4 < 1
and, -pi/4 > -1
therefore,
4.
To know the properties of logarithmic function (click here)
Therefore,
1. for to be defined
x > 0
or,
2. for to be defined
see the above graph,
at x = 1,
therefore, or
therefore, take the intersection of (1) and (2)
or,
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