The function f (x) = (1 / 3) ^ x, X belongs to [- 1,1], the minimum value of function g (x) = f (x) ^ 2-2af (x) + 3 is h (a), find H (a)?

The function f (x) = (1 / 3) ^ x, X belongs to [- 1,1], the minimum value of function g (x) = f (x) ^ 2-2af (x) + 3 is h (a), find H (a)?

If f (x) = (1 / 3) ^ x, X belongs to [- 1,1], then
1/3≤f(x)≤3
Let t = f (x), 1 / 3 ≤ t ≤ 3
g(x)=t^2-2at+3=(t-a)^2-a^2+3
(1) When a > 3, H (a) = 3 ^ 2-6a + 3 = 12-6a
(2) When 1 / 3 ≤ a ≤ 3, H (a) = 3-A ^ 2
(3)a

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