It is known that f (x) = x ^ 2 + X + 1 (1) Find the analytic expression of F (2x); (2) Find the analytic expression of F [f (2x)]; (3) For any x ∈ R, it is proved that f (- 1 / 2 + x) = f (- 1 / 2-x) always holds

It is known that f (x) = x ^ 2 + X + 1 (1) Find the analytic expression of F (2x); (2) Find the analytic expression of F [f (2x)]; (3) For any x ∈ R, it is proved that f (- 1 / 2 + x) = f (- 1 / 2-x) always holds

F (2x) = (2x) ^ 2 + 2x + 1 = 4x ^ 2 + 2x + 1F (f (2x)) = (4x ^ 2 + 2x + 1) ^ 2 + 4x ^ 2 + 2x + 1 + 1 = 16x ^ 4 + 16x ^ 3 + 16x ^ 2 + 6x + 2 prove: left = (- 1 / 2 + x) ^ 2 + (- 1 / 2 + x) + 1 = (3 / 4) + x ^ 2 right = (- 1 / 2-x) ^ 2 + (- 1 / 2-x) + 1 = 1 / 4 + X + X ^ 2-1 / 2-x + 1 = (3 / 4) + x ^ 2 in conclusion: left = right