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Câu 1 trang 14 SGK Đại số và Giải tích 11 Nâng cao

Tìm tập xác định của mỗi hàm số sau :
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Tìm tập xác định của mỗi hàm số sau : a. \(y = \sqrt {3 - \sin x} \) ; b.  \(y = {{1 - \cos x} \over {\sin x}}\) c. \(y = \sqrt {{{1 - \sin x} \over {1 + \cos x}}} \) d. \(y = \tan \left( {2x + {\pi \over 3}} \right)\)

LG a

\(y = \sqrt {3 - \sin x} \) ;

Phương pháp giải:

Biểu thức \(\sqrt P \) có nghĩa khi \(P\ge 0\). Sử dụng đánh giá \(-1 ≤ \sin x ≤ 1\).

Lời giải chi tiết:

Vì \(-1 ≤ \sin x ≤ 1\) nên:

\(\begin{array}{l}
\Rightarrow 1 \ge - \sin x \ge - 1\\
\Rightarrow 1 + 3 \ge - \sin x + 3 \ge - 1 + 3\\
\Rightarrow 4 \ge 3 - \sin x \ge 2 > 0\\
\Rightarrow 3 - \sin x > 0,\forall x \in R
\end{array}\)

Vậy tập xác định của hàm số là: \(D =\mathbb R\)

LG b

\(y = {{1 - \cos x} \over {\sin x}}\)

Phương pháp giải:

Biểu thức \(\frac{P}{Q}\) có nghĩa khi \(Q\ne 0\)

Lời giải chi tiết:

\(y = {{1 - \cos x} \over {\sin x}}\) xác định khi và chỉ khi \(\sin x ≠ 0\)
\(⇔ x ≠ kπ, k \in\mathbb Z\)

Vậy tập xác định \(D =\mathbb R \backslash \left\{ kπ , k \in \mathbb Z\right\}\)

LG c

\(y = \sqrt {{{1 - \sin x} \over {1 + \cos x}}} \)

Phương pháp giải:

Biểu thức \(\sqrt {\frac{P}{Q}} \) xác định khi 

\(\left\{ \begin{array}{l}
\frac{P}{Q} \ge 0\\
Q \ne 0
\end{array} \right.\)

Lời giải chi tiết:

ĐK: \(\left\{ \begin{array}{l}\frac{{1 - \sin x}}{{1 + \cos x}} \ge 0\\1 + \cos x \ne 0\end{array} \right.\left( * \right)\) Ta có: \( - 1 \le \sin x \le 1 \Rightarrow 1 - \sin x \ge 0\) với mọi \(x\). \( - 1 \le \cos x \le 1 \Rightarrow 1 + \cos x \ge 0\) với mọi \(x\). \( \Rightarrow \frac{{1 + \sin x}}{{1 + \cos x}} \ge 0\) với mọi \(x\). Do đó \(\left( * \right) \Leftrightarrow 1 + \cos x \ne 0\) \( \Leftrightarrow \cos x \ne  - 1 \Leftrightarrow x \ne \pi  + k2\pi \) Vậy tập xác định \(D =\mathbb R\backslash\left\{ π + k2π , k \in\mathbb Z\right\}\)

LG d

\(y = \tan \left( {2x + {\pi \over 3}} \right)\)

Phương pháp giải:

Hàm số \(y = \tan u\) xác định khi và chỉ khi \(u \ne \frac{\pi }{2} + k\pi \)

Lời giải chi tiết:

\(y = \tan \left( {2x + {\pi \over 3}} \right)\) xác định ⇔  \(\cos \left( {2x + {\pi \over 3}} \right) \ne 0\)  \( \Leftrightarrow 2x + {\pi \over 3} \ne {\pi \over 2} + k\pi\) \( \Leftrightarrow 2x \ne \frac{\pi }{6} + k\pi \) \(\Leftrightarrow x\ne {\pi \over {12}} + k{\pi \over 2},k \in \mathbb Z\) Vậy tập xác định \(D =\mathbb R\backslash \left\{ {{\pi \over {12}} + k{\pi \over 2},k \in\mathbb Z} \right\}\)

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