Решение 1. Область определения y = 2cos(x-п/3) D(y) = R 2. Область значения - 1 ≤ 2cos(x-п/3) ≤ 1 - 1/2 ≤ cos(x-п/3) ≤ 1/2 1) cos(x-п/3) ≥ - 1/2 - arccos(-1/2) + 2πk ≤ x - п/3 ≤ arccos(-1/2) + 2πk, k ∈ Z - 2π/3 + 2πk ≤ x - п/3 ≤ 2π/3 + 2πk, k ∈ Z - 2π/3 + π/3 + 2πk ≤ x ≤ 2π/3 + π/3 + 2πk, k ∈ Z - π/3 + 2πk ≤ x ≤ π + 2πk, k ∈ Z 2) cos(x-п/3) ≤ - 1/2 arccos(-1/2) + 2πk ≤ x - п/3 ≤ 2π - arccos(-1/2) + 2πk, k ∈ Z 2π/3 + 2πk ≤ x - п/3 ≤ 2π - 2π/3 + 2πk, k ∈ Z 2π/3 + 2πk ≤ x - п/3 ≤ 4π/3 + 2πk, k ∈ Z 2π/3 + π/3 + 2πk ≤ x ≤ 4π/3 + π/3 + 2πk, k ∈ Z π + 2πk ≤ x ≤ 5π/3 + 2πk, k ∈ Z
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y=(x+2)² - 4 * * * парабола с вершиной в точке ( -2 ; - 4) * * *
а)
ООФ : x∈(-∞; ∞) * * * x∈ R * * *
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б)
y =0 * * * (x+2)² - 4=x²+4x+4 -4 = x²+4x =x(x+4) * * *
y=(x+2)² - 4 =(x+2)² -2² =(x+2 -2)(x+2+2) =(x+4)x ;
y=0 ⇒ x₁ =- 4 ; x₂=0 .
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в)
y < 0 ; (x+4)x < 0 ⇒ x ∈(- 4 ; 0) .
+ - +
--------------- [ - 4 ] --------- [ 0] ---------------
y > 0 ⇒ x ∈(-∞; - 4) ∪ (0 ; ∞) .
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г)
функция возрастает (↑), если x∈[ -2 ; ∞) ,
убывает (↓),если ∈ ( -∞ ; -2] .
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д)
E(y) = [ -4 ;∞).
* * * Все это можно выполнить с производной * * *
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5.
а) y =√(3-x) + Log(3) (x² -1) .
ООФ : {3- x ≥ 0 ; {x ≤ 3 ; {x ∈ (-∞; 3] ;
{ x² -1 >0. {(x+1)(x-1) >0 . { x∈(-∞; -1) ∪(1 ; ∞) .
///////////////////////////////////////[3] ----------
/////////// (-1) -----(1) /////////////////////////////
ответ : x ∈ (-∞; -1) ∪ (1 ; 3] .
б) y= корень все под корень 4/ 1/x^2 - 1
y =√ 4/(1/x² -1)
ООФ : 1/x² -1 >0 ⇔(1-x²) /x² > 0 ⇔ (1+x)(1-x) /x² >0
- + + -
-----------------(-1) /////////// (0) ///////////// (1) ----------
ответ : x ∈( -1 ; 0) ∪(0 ;1) .
1. Область определения
y = 2cos(x-п/3)
D(y) = R
2. Область значения
- 1 ≤ 2cos(x-п/3) ≤ 1
- 1/2 ≤ cos(x-п/3) ≤ 1/2
1) cos(x-п/3) ≥ - 1/2
- arccos(-1/2) + 2πk ≤ x - п/3 ≤ arccos(-1/2) + 2πk, k ∈ Z
- 2π/3 + 2πk ≤ x - п/3 ≤ 2π/3 + 2πk, k ∈ Z
- 2π/3 + π/3 + 2πk ≤ x ≤ 2π/3 + π/3 + 2πk, k ∈ Z
- π/3 + 2πk ≤ x ≤ π + 2πk, k ∈ Z
2) cos(x-п/3) ≤ - 1/2
arccos(-1/2) + 2πk ≤ x - п/3 ≤ 2π - arccos(-1/2) + 2πk, k ∈ Z
2π/3 + 2πk ≤ x - п/3 ≤ 2π - 2π/3 + 2πk, k ∈ Z
2π/3 + 2πk ≤ x - п/3 ≤ 4π/3 + 2πk, k ∈ Z
2π/3 + π/3 + 2πk ≤ x ≤ 4π/3 + π/3 + 2πk, k ∈ Z
π + 2πk ≤ x ≤ 5π/3 + 2πk, k ∈ Z