Properties of Greatest Integer Function `(i)[-n]=-[n] (ii)[x+k]=[x]+k (iii) [-x]=-[x]-1 (iv) [x]+[-x]={-1 ; xnotinZ ; 0 ; xepsilonZ} (iv) [x]-[-x]={2[x]+1 ; if xnotinZ ; 2[x] ; if xepsilonZ`

Properties of smallest integer function property (i)[-n]=-[n]where n epsilon N (ii) [-x]=-[x]+1; where xepsilon R-Z (iii) [x+n]=[x]+n where x epsilon R-Z and n epsilon Z (iv) [x]+[-x]={1; x notinZ ; 0 ; xinZ} (v) [x]-[-x]={2[x]-1; xnotinZ ; 2[x] ; if xinZ

Properties of Greatest Integer Function (vi)[x]>=k rArr x>=k; whereke Z(vii)[x] k rArr x>(k+1) ; where k varepsilon Z

If [.] denotes the greatest integer function, then f(x)=[x]+[x+(1)/(2)]

Prove that the greatest integer function defined by f(x)=[x],0

Find the value of the following expressions, when x=-1 (i) 2x-7( ii) -x+2 (iii) x^(2)+2x+1 (iv) 2x^(2)-x-2

Find the domain of functions : (i) f(x)= 1/sqrt(1-cosx) (ii) (x^3-x+3)/(x^2-1) (iii) (3x)/(2x-8) (iv) x|x|

Solve for x (where [*] denotes greatest integer function and {*} represent fractional part function) (i) [2x]=1 (ii) {x}^(2)+[x]=2 (iii) 6{x}^(2)-5{x}+1=0 (iv) 6{x}^(2)-5{x}-1=0

sin^(2) x+2sin x cos x-3cos^(2) x = 0 if (i) tan x = 3 (ii) tan x = -1 (iii) x = n pi+(pi)/(4) , n in I (iv) x = n pi+tan^(-1) (3), n in I

Find the derivative of the following functions from first principle. (i) x^3-27 (ii) (x" "" "1)" "(x" "" "2) (iii) 1/(x^2) (iv) (x+1)/(x-1)