Why does the monotone increasing interval of function y = sin (π / 3-1 / 2x), X ∈ [- 2 π, 2 π] have union,

Why does the monotone increasing interval of function y = sin (π / 3-1 / 2x), X ∈ [- 2 π, 2 π] have union,

∵y＝sin(π/3-1/2X)
∴y＝﹣sin(1/2X－π/3)
∵2kπ＋π／2≤1/2X－π/3≤2kπ＋3π／2
∴4kπ＋5π／3≤x≤4kπ＋11π／3
And ∵ x ∈ [- 2 π, 2 π]
When k = - 1, - 2 π≤ x ≤ - π / 3
When k = 0, 5 π / 3 ≤ x ≤ 11 π / 3
So the monotone increasing intervals of y = sin (π / 3-1 / 2x) x ∈ [- 2 π, 2 π] are [- 2 π, - π / 3] and [5 π / 3,11 π / 3]
Note: a function has two or more monotone intervals and cannot be written as a union

Monotone increasing interval of function f (x) = sin (- 2x),

y=sin(-2x)=-sin2x
T = 2x is an increasing function
y=- sint
So the increasing interval of the function is required,
Then y = - Sint is a decreasing function
So 2K π + π / 2 ≤ t ≤ 2K π + 3 π / 2
That is 2K π + π / 2 ≤ 2x ≤ 2K π + 3 π / 2
kπ+π/4≤ x≤2kπ+3π/4
Therefore, the increasing interval is [K π + π / 4, K π + 3 π / 4], K ∈ Z

The monotone interval of function f (x) = x ^ 2 + 4x + 5 / x ^ 2 + 4x + 4 is pointed out, and the size of F (- 3.14) and f (- 0.7) are compared

F (x) = x ^ 2 + 4x + 5 / x ^ 2 + 4x + 4 = 1 + 1 / x ^ 2 + 4x + 4 = 1 + 1 / (x + 2) ^ 2, f (x) is a decreasing function on (- ∞, - 2), (- 2, + ∞), f (- 3.14) = 1 + 1 / (- 3.14 + 2) ^ 2, = 1 + 1 / 1.14 ^ 2, f (- 0.7) = 1 + 1 / (- 0.7 + 2) ^ 2, = 1 + 1 / 1.3 ^ 2, because 1.14 ^ 2F (- 0.7)