Domain of given function

Aditi Bhatnagar

The function f(X)=sin^-1(x-x²)+ (1-(1/ IXI))^1/2+1/[x²-1] is defined in which interval ? ([.] is greatest integer)

Manasi

domain of sin^-1(x-x²)

-1 =< x-x²< = 1

x² - x - 1 =< 0

=> ( x-1/2 )² - (5/4) =< 0

=> [ x - {(1-sqrt5)/2} ] [ x - {(1+sqrt5)/2} ] =< 0

=> [ x - (- 0.6 ) ] [ x - 1.6 ] =< 0

=> x

[ -0.6, 1.6] .....................(A)

domain of (1-(1/ IXI))^1/2

firstly, IxI should not be 0 => x (not =) 0 .................(i)

1-(1/ IxI) > = 0

=> 1 > = (1/ IxI)

=> |x| > = 1

=> x =< -1 or x >= 1 ..........................(ii)

combining (i) and (ii), domain : x (-infinity, -1] [1, infinity) .............................(B)

domain of 1/[x²-1]

[x²-1] (not =) 0

=> (x²-1)<0 or, (x²-1) >=1

(x²-1) < 0

=> x

(-1,1) .............................................(iii)

(x²-1) > = 1

=> x

(-infinity , -sqrt2 ]

[sqrt2, infinity ) ..........................(iv)

combining (iii) and (iv), x (-infinity , -sqrt2 ] (-1,1) [sqrt2, infinity ) .....................(C)

as all the three conditions given by (A), (B) and (C), should be fullfilled, the domain of the entire expression can be found out by taking the intersection of the three :)

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