Let f: R S be an onto function defined by f(x) = [ 3x^2 - 2x + 3 x^2 + x + 1 ] , where [t] denotes the greatest integer less than or equal to t . The number of elements in the set S is :

Step-by-step solution

Let y = 3x^2 - 2x + 3 x^2 + x + 1 . Rearranging the equation to form a quadratic in x : y(x^2 + x + 1) = 3x^2 - 2x + 3 (y - 3)x^2 + (y + 2)x + (y - 3) = 0 Since x R , the discriminant of this quadratic equation must be non-negative ( D 0 ): (y + 2)^2 - 4(y - 3)^2 0 Using the difference of squares a^2 - b^2 = (a - b)(a + b) : ((y + 2) - 2(y - 3))((y + 2) + 2(y - 3)) 0 (y + 2 - 2y + 6)(y + 2 + 2y - 6) 0 (8 - y)(3y - 4) 0 Multiplying by -1 reverses the inequality: (y - 8)(3y - 4) 0 This gives y [ 4 3 , 8 ] . Since x^2