Find the number of solutions for x - x = sin x , where · denotes the greatest integer function.

Consider two functions, y = [ [ x ] - x ] and y = sin x y = [ [ x ] - x ] = [ - { x } ] ( ∵ x = [ x ] + { x } ) We know that 0 ≤ { x } < 1 for x ∈ R ⇒ - 1 < - { x } ≤ 0 ∀ x ∈ R ⇒ - x = 0 , x ∈ I - 1 , x ∉ I So the graph of both functions is As we can see that in the figure, both curves have an infinite number of intersection points. So, the number of solutions of [ [ x ] - x ] = sin x are infinite.

Find the number of solutions for x - x = sin x , where · denotes the greatest integer function.

Consider two functions, y = [ [ x ] - x ] and y = sin x y = [ [ x ] - x ] = [ - { x } ] ( ∵ x = [ x ] + { x } ) We know that 0 ≤ { x } < 1 for x ∈ R ⇒ - 1 < - { x } ≤ 0 ∀ x ∈ R ⇒ - x = 0 , x ∈ I - 1 , x ∉ I So the graph of both functions is As we can see that in the figure, both curves have an infinite number of intersection points. So, the number of solutions of [ [ x ] - x ] = sin x are infinite.

Find the values of x graphically, which satisfy x 2 x - 1 ≤ 1 .

Consider two functions, y = x 2 and y = | x - 1 | Case-I: If x ≥ 1 For the intersection points of both curves, x 2 = x - 1 ⇒ x 2 - x + 1 = 0 Here, discriminant of the quadratic, D = 1 - 4 < 0 . Hence, no solution or no points of intersection. Case-II: If x < 1 For the intersection points of both curves, x 2 = - ( x - 1 ) ⇒ x 2 + x - 1 = 0 ⇒ x = - 1 + 5 2 and - 1 - 5 2 Graphs of both curves is As shown in figure, x 2 ≤ | x - 1 | ∀ x ∈ - 1 - 5 2 , - 1 + 5 2 ⇒ x 2 x - 1 ≤ 1 ∀ x ∈ - 1 - 5 2 , - 1 + 5 2

Find the number of value(s) of x for which x 3 - x = 3 , where · denotes the greatest integer function.

Consider two curves, y = x 3 and y = [ x ] + 3 We know that domain and range of a cubic function is R . Now domain of x is R and range is Z . We can get the graph of x + 3 by vertically shifting upward the graph of x by 3 units. Combined graph of both curves is As we can see in the figure, both curves have only 1 intersection point. So, the number of solution of x 3 - 3 = [ x ] is 1 .