a: ĐKXĐ: x\(\in\)R\{3}

b: ĐKXĐ: \(\left\{{}\begin{matrix}x>1\\x\ne2\end{matrix}\right.\)

1. Hàm số xác định `<=> 1-cosx \ne 0<=>cosx \ne 1<=>x \ne k2π`

Vì: `1+cosx >=0 forallx ; 1-cosx >=0 forall x`

2. Hàm số xác định `<=> sin^2x \ne cos^2x <=> (1-cos2x)/2 \ne (1+cos2x)/2`

`<=>cos2x \ne 0<=> 2x \ne π/2+kπ <=> x \ne π/4+kπ/2`

3. Hàm số xác định `<=> cos2x \ne 0<=> x \ne π/4+kπ/2 (k \in ZZ)`.

ĐKXĐ:

\(\left\{{}\begin{matrix}x+1\ge0\\x^2-2\ge0\\5-x>0\\x^2-2x-3\ne0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ge-1\\\left|x\right|\ge\sqrt{2}\\x< 5\\x\ne-1;x\ne3\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}\sqrt{2}\le x< 5\\x\ne3\end{matrix}\right.\)

a, y xác định `<=> 3cos(2x+3) \ne 0`

`<=>cos(2x+3) \ne 0`

`<=>2x+3 \ne π/2+kπ`

`<=>x \ne π/4 -3/2 +k π/2 (k \in ZZ)`

b, y xác định `<=> sin(x/3+π/4) \ne0`

`<=> x/3+π/4 \ne kπ`

`<=> x \ne (-3π)/4+ k3π`

ĐKXĐ: 

a.

\(cos\left(2x+3\right)\ne0\)

\(\Leftrightarrow2x+3\ne\dfrac{\pi}{2}+k\pi\)

\(\Leftrightarrow x=-\dfrac{3}{2}+\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)

b.

\(sin\left(\dfrac{x}{3}+\dfrac{\pi}{4}\right)\ne0\)

\(\Leftrightarrow\dfrac{x}{3}+\dfrac{\pi}{4}\ne k\pi\)

\(\Leftrightarrow x\ne-\dfrac{3\pi}{4}+k3\pi\)

Hàm số xác định khi |2x - 3| ≥ 0

Mệnh đề trên đúng với ∀ x ∈ R

Vậy tập xác định là D = R

\(y\) có TXĐ là \(\mathbb{R}\) \(\Leftrightarrow (mx+3)(x-2) ≥0\)

TH1: \(\left[ \begin{array}{l}mx+3\\x-2=0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x=\dfrac{-3}{m} (m\ne0)\\x=2\end{array} \right.\)

TH2: \(\begin{cases}mx+3>0\\x-2>0\\\end{cases} \Leftrightarrow \begin{cases}x > \dfrac{-3}{m} \\x>2\\\end{cases} \)

TH3: \(\begin{cases}mx+3<0\\x-2<0\\\end{cases} \Leftrightarrow \begin{cases}x < \dfrac{-3}{m}\\x<2\\\end{cases} \)

Vậy...

\(\text{1) Đ}K:\left\{{}\begin{matrix}sinx\ne0\\1-sinx\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ne m\pi\\x\ne\frac{\pi}{2}+n2\pi\end{matrix}\right.\)

\(2\text{) }ĐK:\left\{{}\begin{matrix}cos\left(2x+\frac{\pi}{3}\right)\ne0\\sinx\ne0\end{matrix}\right.\Leftrightarrow\\ \left\{{}\begin{matrix}2x+\frac{\pi}{3}\ne\frac{\pi}{2}+m\pi\\x\ne n\pi\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ne\frac{\pi}{12}+\frac{m\pi}{2}\\x\ne n\pi\end{matrix}\right.\)

\(3\text{) }ĐK:\left\{{}\begin{matrix}\frac{5-3cos2x}{1+sin\left(2x-\frac{\pi}{2}\right)}\ge0\\1+sin\left(2x-\frac{\pi}{2}\right)\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5-3cos2x\ge0\\sin\left(2x-\frac{\pi}{2}\right)\ne-1\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}cos2x\le\frac{5}{3}\left(T/m\right)\\2x-\frac{\pi}{2}\ne\frac{3\pi}{2}+k2\pi\end{matrix}\right.\Leftrightarrow x\ne\pi+k\pi\)

\(4\text{) }ĐK:\left\{{}\begin{matrix}sin\left(x+\frac{\pi}{3}\right)\ne0\\cos\left(3x-\frac{\pi}{4}\right)\ne0\\tan\left(3x-\frac{\pi}{4}\right)\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+\frac{\pi}{3}\ne a\pi\\3x-\frac{\pi}{4}\ne\frac{\pi}{2}+b\pi\\3x-\frac{\pi}{4}\ne c\pi\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ne-\frac{\pi}{3}+a\pi\\x\ne\frac{\pi}{4}+\frac{b\pi}{3}\\x\ne\frac{\pi}{12}+\frac{c\pi}{3}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ne-\frac{\pi}{3}+a\pi\\x\ne\frac{\pi}{12}+\frac{k\pi}{6}\end{matrix}\right.\)

a.

Miền xác định của hàm số là miền đối xứng: \(D=R\backslash\left\{0\right\}\)

\(f\left(-x\right)=\frac{\left|6-\left(-2x\right)\right|-\left|6+\left(-2x\right)\right|}{\left(-x\right)^2}=-\frac{\left|6-2x\right|-\left|6+2x\right|}{x^2}=-f\left(x\right)\)

Hàm lẻ

b.

ĐKXĐ: \(\left\{{}\begin{matrix}6-3x\ge0\\x\ne0\\x+1>0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\le2\\x\ne0\\x>-1\end{matrix}\right.\)

\(D=\left(-1;0\right)\cup(0;2]\)

a/ ĐKXĐ: \(cos\left(2x+\frac{\pi}{3}\right)-1\ne0\)

\(\Leftrightarrow cos\left(2x+\frac{\pi}{3}\right)\ne1\)

\(\Leftrightarrow2x+\frac{\pi}{3}\ne k2\pi\)

\(\Leftrightarrow x\ne-\frac{\pi}{6}+k\pi\)

b/ ĐKXĐ: \(tanx+1\ne0\Leftrightarrow tanx\ne-1\)

\(\Leftrightarrow x\ne-\frac{\pi}{4}+k\pi\)

c/ \(1-sin2x\ge0\Leftrightarrow1\ge sin2x\) (luôn đúng)

\(\Rightarrow\) Hàm số xác định với mọi x thuộc R