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Câu 23 trang 31 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:

LG a

\(y = {{1 - \cos x} \over {2\sin x + \sqrt 2 }}\)

Lời giải chi tiết:

\(y = {{1 - \cos x} \over {2\sin x + \sqrt 2 }}\) xác định  \( \Leftrightarrow 2\sin x + \sqrt 2 \ne 0\) \( \Leftrightarrow \sin x \ne - {{\sqrt 2 } \over 2} \) \(\Leftrightarrow \left\{ {\matrix{{x \ne - {\pi \over 4} + k2\pi } \cr {x \ne {{5\pi } \over 4} + k2\pi } \cr} } \right.\) Vậy tập xác định của hàm số đã cho là : \(D =\mathbb R \backslash  \left( {\left\{ { - {\pi \over 4} + k2\pi ,k \in\mathbb Z} \right\} \cup \left\{ {{{5\pi } \over 4} + k2\pi ,k \in\mathbb Z} \right\}} \right)\)

LG b

\(y = {{\sin \left( {x - 2} \right)} \over {\cos 2x - \cos x}}\)

Lời giải chi tiết:

\(y = {{\sin \left( {x - 2} \right)} \over {\cos 2x - \cos x}}\) xác định \( \Leftrightarrow \cos 2x - \cos x \ne 0\) \(\eqalign{& \Leftrightarrow \cos 2x \ne \cos x \cr & \Leftrightarrow \left\{ {\matrix{{2x \ne x + k2\pi } \cr {2x \ne - x + k2\pi } \cr} } \right. \cr&\Leftrightarrow \left\{ {\matrix{{x \ne k2\pi } \cr {x \ne k{{2\pi } \over 3}} \cr} } \right. \cr&\Leftrightarrow x \ne k{{2\pi } \over 3} \cr} \)  Vậy \(D =\mathbb R \backslash  \left\{ {k{{2\pi } \over 3},k \in\mathbb Z} \right\}\)

LG c

\(y = {{\tan x} \over {1 + \tan x}}\)

Lời giải chi tiết:

\(y = {{\tan x} \over {1 + \tan x}}\) xác định

\( \Leftrightarrow \left\{ \begin{array}{l}
x \ne \frac{\pi }{2} + k\pi \\
1 + \tan x \ne 0
\end{array} \right. \) \(\Leftrightarrow \left\{ \begin{array}{l}
x \ne \frac{\pi }{2} + k\pi \\
\tan x \ne - 1
\end{array} \right.\)

\(\Leftrightarrow \left\{ {\matrix{{x \ne {\pi \over 2} + k\pi } \cr {x \ne - {\pi \over 4} + k\pi } \cr} } \right.\) Vậy  \(D =\mathbb R \backslash  \left( {\left\{ {{\pi \over 2} + k\pi ,k \in\mathbb Z} \right\} \cup \left\{ { - {\pi \over 4} + k\pi ,k \in\mathbb Z} \right\}} \right)\)

Chú ý:

Một số em thường quên mất điều kiện để \(\tan x\) xác định, đó là \(x \ne \frac{\pi }{2} + k\pi \) dẫn đến thiếu điều kiện.

LG d

\(y = {1 \over {\sqrt 3 \cot 2x + 1}}\)

Lời giải chi tiết:

\(y = {1 \over {\sqrt 3 \cot 2x + 1}}\) xác định

\( \Leftrightarrow \left\{ \begin{array}{l}
2x \ne k\pi \\
\sqrt 3 \cot 2x + 1 \ne 0
\end{array} \right. \) \(\Leftrightarrow \left\{ \begin{array}{l}
x \ne \frac{{k\pi }}{2}\\
\cot 2x \ne - \frac{1}{{\sqrt 3 }}
\end{array} \right.\) \( \Leftrightarrow \left\{ \begin{array}{l}
x \ne \frac{{k\pi }}{2}\\
2x \ne - \frac{\pi }{3} + k\pi
\end{array} \right. \) \(\Leftrightarrow \left\{ \begin{array}{l}
x \ne \frac{{k\pi }}{2}\\
x \ne - \frac{\pi }{6} + \frac{{k\pi }}{2}
\end{array} \right.\)

Vậy \(D =\mathbb R \backslash  \left( {\left\{ {k{\pi \over 2},k \in\mathbb Z} \right\} \cup \left\{ { - {\pi \over 6} + k{\pi \over 2},k \in\mathbb Z} \right\}} \right)\)

Chú ý:

Một số em thường quên mất điều kiện để \(\cot 2x\) xác định, đó là \(2x \ne k\pi \) dẫn đến thiếu điều kiện.

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