# 1. If [x]^2 + (x)^2 >25 where [.] is greatest integer function and (.) is least integer functin den x belongs to ?2. domain of function f(x)=1/underroot(4x-|x^2 - 10x +9|).here |.| is modulus function.

Swapnil Saxena
102 Points
12 years ago

Since root is defined only for >=0 and 1/x is defined for not= 0 thus 4x-|x^2 - 10x +9|> 0

However 4x-|x2 - 10x +9| can take two forms

Case I : For (x2 - 10x +9) >0  ==>  (x-1)(x-9)>0 , it is equal to 4x-(x2 - 10x +9) =>  -x2+14x-9

Case II : For (x2 - 10x +9) <0  ==> (x-1)(x-9)<0  it is equal to 4x+(x2 - 10x +9) ==> x2-6x+9

Evaluating case I :

Now we ve to find when  (x-1)(x-9)>0 ,so that the equation takes form   -x2+14x-9

For this, either (x>1 and x>9 ==> x>9) or (x< 1 and x< 9 ==> x<1_

So for (-infinity, 1) and (9, infinity) , ths expression can take form -x2+14x-9,

--------------------------------------------------------------------------------------------------

In such a situation, finding when   -x2+14x-9 > 0 ,

Using Wavy curvy method, x is (7-2root(10),7+2root(10))

So using intersection of the two domains , between (7-2root(10),1)U(9,7+2root(10)),

4x-|x^2 - 10x +9|> 0

Evaluating case II:

Now we ve to find when  (x-1)(x-9)<0 ,

for this purpose (x<1) and (x>9) , not possible

for this purpose (x>1) and (x<9) ,thus for (1,9), the equation will take form x2-6x+9

Now we have to evaluate when  x2-6x+9 >0,

Again using wavy curvy, x is always positive for this equation

So x can belong to (1,9)

---------------------------------------------------------------------------------------------------

After Evaluating both the case we find that domian can be (7-2root(10),1) U (1,9) U (9,7+2root(10)) =

( 7-2root(10) , 7+2root(10))

Swapnil Saxena
102 Points
12 years ago

The domain of the function is (-infinity,-3) U (4,infinity)

The solution is as follows:

For making a rough estimate when [x]2 + (x)2 >25

Since the least value that [x] can take is x-1

Suppose for any no. ,  (x-1(approx of 0.9999))2 + x2 > 25 .

Solving the equation x> 4 or x<-3

Thus for (-infinity,-3) U (4,infinity) is the domain of the function

Swapnil Saxena
102 Points
12 years ago

Sorry for a slight mistake  (7-2root(10),1) U (1,9) U (9,7+2root(10)) is not equal to to (7-2root(10),7+2root(10)) but is equal to  (7-2root(10),7+2root(10)) excluding 1 and 9

manuj mittal
4 Points
12 years ago

in d first question put x=4 it is satisfying d equation and put -3.1 it is not satisfying d equation..........actually i know how to solve dis problem but problem is dis my answer is not matching with answer of book.my answer is (-infinity,-4) U [4,infinity) and book ans is (-infinity,-4] U [4,infinity).Clearly x=-4 will satisfy d equation but i m not getting it mathematically........U CAN SOLVE DIS BY PUTTING X= I + F WHERE I=INTEGER AND F= FRACTIONAL PART FUNCTION.NOW TRY TO SOLVE IT ........I HOPE MY AND UR ANSWERS WILL MATCH.

manuj mittal
4 Points
12 years ago

thanks 4 d answer of second question........and sryyy by mistake i disapproved it........i was disapproving d answer of first question.........

manuj mittal
4 Points
12 years ago

once again i m sryy...... ur answer of 2nd question is also wrong it is (7-root(40),7+root(40)) -{3}. but thanks 4 efforts. with d help of ur solution i understood how d correct answer is coming.......u did some mistakes in ur solution......

Swapnil Saxena
102 Points
12 years ago

No need to say sorry !!! i m srry for my mistake. my solution is incorrect. but finally i m relieved u got the answer.