Let D = x ∈ ℝ : f x = x - | x | x - [ x ] is defined and C be the range of the real function g x = 2 x 4 + x 2 . Then D ∩ C =

Step-by-step solution

The given function is f x = x - | x | x - [ x ] Now, x - | x |   ≥   0   &   x - [ x ] > 0 ⇒ x ≥ | x | and x > [ x ] Therefore, x   ∈ ℝ + - (all integers) Again, g x = 2 x 4 + x 2 ⇒ y = 2 x 4 + x 2 Rewrite the above equation. y x 2 - 2 x + 4 y = 0 ⇒ x = 2 ± 4 - 16 y 2 2 y From the above equation 4 - 16 y 2 ≥ 0   &   y ≠ 0 1 - 4 y 2 ≥ 0   &   y ≠ 0 Therefore, y ∈ - 1 2 , 1 2 -