Задача №1026

Решение

Пункт №1

\( \left\{\begin{aligned} & 1-x\gt{0};\\ & 1-x\neq{1};\\ & x+2\ge{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\lt{1};\\ & x\neq{0};\\ & x\ge{-2}. \end{aligned}\right. \)

Область определения: \(D(y)=[-2;0)\cup(0;1)\).

Пункт №2

\( \left\{\begin{aligned} & 3-x\ge{0};\\ & -1\le\frac{3-2x}{5}\le{1}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\le{3}\\ & -1\le{x}\le{4}. \end{aligned}\right. \Leftrightarrow -1\le{x}\le{3}. \)

Область определения: \(D(y)=[-1;3]\).

Пункт №3

\( \left\{\begin{aligned} & 4-x\gt{0};\\ & -1\le\frac{x-3}{2}\le{1}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\lt{4}\\ & 1\le{x}\le{5}. \end{aligned}\right. \Leftrightarrow 1\le{x}\lt{4}. \)

Область определения: \(D(y)=[1;4)\).

Пункт №4

\( \left\{\begin{aligned} & x\ge{0};\\ & x-2\neq{0};\\ & 2x-3\gt{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\neq{2};\\ & x\gt\frac{3}{2}. \end{aligned}\right. \)

Область определения: \(D(y)=\left(\frac{3}{2};2\right)\cup(2;+\infty)\).

Пункт №5

\( \left\{\begin{aligned} & x-1\ge{0};\\ & 1-x\ge{0}. \end{aligned}\right. \Leftrightarrow x=1. \)

Область определения: \(D(y)=\{1\}\).

Пункт №6

\( \left\{\begin{aligned} & 4-x^2\neq{0};\\ & x^3-x\gt{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\neq{-2};\\ & x\neq{2};\\ & x(x-1)(x+1)\gt{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\neq{2};\\ & -1\lt{x}\lt{0};\\ & x\gt{1}. \end{aligned}\right. \)

Область определения: \(D(y)=(-1;0)\cup(1;2)\cup(2;+\infty)\).

Пункт №7

\( \left\{\begin{aligned} & \sin(x-3)\gt{0};\\ & 16-x^2\ge{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & 3+2\pi{n}\lt{x}\lt{3+\pi+2\pi{n}},\;n\in{Z};\\ & -4\le{x}\le{4}. \end{aligned}\right. \Leftrightarrow \left[\begin{aligned} &3-2\pi\lt{x}\lt{3-\pi};\\ & 3\lt{x}\le{4}. \end{aligned}\right. \)

Область определения: \(D(y)=(3-2\pi;3-\pi)\cup(3;4]\).

Пункт №8

\( \left\{\begin{aligned} & \sin{x}\ge{0};\\ & 16-x^2\ge{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & 2\pi{n}\le{x}\le{\pi+2\pi{n}},\;n\in{Z};\\ & -4\le{x}\le{4}. \end{aligned}\right. \Leftrightarrow \left[\begin{aligned} &-4\le{x}\le{-\pi};\\ & 0\le{x}\le\pi. \end{aligned}\right. \)

Область определения: \(D(y)=[-4;-\pi]\cup[0;\pi]\).

Пункт №9

\( \sin{x}\gt{0} \Leftrightarrow 2\pi{n}\lt{x}\lt{\pi+2\pi{n}},\;n\in{Z}. \)

Область определения: \(D(y)=\bigcup\limits_{n\in{Z}}(2\pi{n};\pi+2\pi{n})\).

Пункт №10

\( \frac{x-5}{x^2-10x+24}\gt{0} \Leftrightarrow (x-4)(x-5)(x-6)\gt{0}. \)

Область определения: \(D(y)=(4;5)\cup(6;+\infty)\).

Пункт №11

\( \left\{\begin{aligned} & \frac{x-2}{x+2}\ge{0};\\ & x+2\neq{0};\\ & \frac{1-x}{\sqrt{1+x}}\ge{0};\\ & 1+x\gt{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & \left[\begin{aligned}&x\ge{2};\\&x\lt{-2}.\end{aligned}\right.;\\ & -1\lt{x}\le{1}. \end{aligned}\right. \)

Область определения: \(D(y)=\varnothing\).

Пункт №12

\( \left\{\begin{aligned} & x^2-3x+2\ge{0};\\ & 3+2x-x^2\gt{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & \left[\begin{aligned}&x\le{1};\\&x\ge{2}.\end{aligned}\right.;\\ & -1\lt{x}\lt{3}. \end{aligned}\right. \)

Область определения: \(D(y)=(-1;1]\cup[2;3)\).

Пункт №13

\( x^2+x+1 =\left(x+\frac{1}{2}\right)^2+\frac{3}{4} \gt{0} \)

Так как при любом \(x\in{R}\) имеем \(x^2+x+1\gt{0}\), то область определения \(D(y)=(-\infty;+\infty)\).

Пункт №14

\( \left\{\begin{aligned} & x-4\ge{0};\\ & 6-x\ge{0};\\ & \sqrt{x-4}+\sqrt{6-x}\gt{0}. \end{aligned}\right. \Leftrightarrow \left\{\begin{aligned} & x\ge{4};\\ & x\le{6}. \end{aligned}\right. \Leftrightarrow 4\le{x}\le{6}. \)

Область определения: \(D(y)=[4;6]\).

Пункт №15

\( \left\{\begin{aligned} & x^2-5x+16\gt{0};\\ & 1-\lg\left(x^2-5x+16\right)\gt{0}. \end{aligned}\right. \Leftrightarrow x^2-5x+6\lt{0} \Leftrightarrow 2\lt{x}\lt{3}. \)

Область определения: \(D(y)=(2;3)\).