Given the function f (x) = LG (x + 1); (1) if 0 ＜ f (1-2x) - f (x) ＜ 1, find the value range of X; (2) if G (x) is an even function with a period of 2, and if 0 ≤ x ≤ 1, G (x) = f (x), find the analytic expression of the function y = g (x) (x ∈ [- 1,1]) | Math enthusiast | Mathematical art

Given the function f (x) = LG (x + 1); (1) if 0 ＜ f (1-2x) - f (x) ＜ 1, find the value range of X; (2) if G (x) is an even function with a period of 2, and if 0 ≤ x ≤ 1, G (x) = f (x), find the analytic expression of the function y = g (x) (x ∈ [- 1,1])

(1) F (1-2x) = LG (2-2x), from 2-2x ＞ x x + 1 ＞ 0, we can get - 1 ＜ x ＜ 1. From 0 ＜ f (1-2x) - f (x) ＜ 1, we can get 0 ＜ lg2-2xx + 1 ＜ 1, 1 ＜ 2-2xx + 1 ＜ 10 & nbsp; & nbsp; ∵ x + 1 ＞ 0, ∵ x + 1 ＜ 2-2x ＜ 10x + 10, ∵ 23 ＜ x ＜ 13. ∵ - 1 ＜ x ＜ 1, ∵ 23 ＜ x ＜ 13. (2) when x ∈ [- 1, 0], X ∈ [0, 1], so y = G (x) = g (- x) = LG [(- x) + 1] = LG (1-x), ∵ function y = g (x) = LG (1-x), X ∈ [- 1, 0) LG (1 + x), X ∈ [0, 1]

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