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Câu 7 trang 16 SGK Đại số và Giải tích 11 Nâng cao

Xét tính chẵn – lẻ của mỗi hàm số sau :
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Xét tính chẵn – lẻ của mỗi hàm số sau : a.  \(y = \cos \left( {x - {\pi \over 4}} \right)\) b.  \(y = \tan \left| x \right|\)  c.  \(y = \tan x - \sin 2x.\)

LG a

\(y = \cos \left( {x - {\pi \over 4}} \right)\)

Lời giải chi tiết:

Ta có:

\(\eqalign{
& f\left( x \right) = \cos \left( {x - {\pi \over 4}} \right)\cr&f\left( {{\pi \over 4}} \right) = 1,f\left( { - {\pi \over 4}} \right) = 0 \cr 
ꦿ& f\left( { - {\pi \over 4}} \right) \ne f\left( {{\pi \over 4}} \right)\cr& \text{và }f\left( { - {\pi \over 4}} \right) \ne - f\left( {{\pi \over 4}} \right) \cr} \)

Nên \(y = \cos \left( {x - {\pi \over 4}} \right)\) không phải là hàm số chẵn cũng không phải là hàm số lẻ.

LG b

\(y = \tan \left| x \right|\) 

Lời giải chi tiết:

\(f(x) = \tan|x|\). Tập xác định \(D =\mathbb R \backslash  \left\{ {{\pi \over 2} + k\pi ,k \in \mathbb Z} \right\}\) \(x \in D ⇒ -x \in D\) và \(f(-x) = \tan |-x| = \tan |x| = f(x)\) Do đó \(y = \tan |x|\) là hàm số chẵn.

LG c

\(y = \tan x - \sin 2x.\)

Lời giải chi tiết:

\(f(x) = \tan x – \sin 2x\). Tập xác định \(D =\mathbb R \backslash \left\{ {{\pi \over 2} + k\pi ,k \in\mathbb Z} \right\}\) \(x \in D ⇒ -x \in D\) và \(f(-x) = \tan(-x) – \sin(-2x)\) \(= -\tan x + \sin 2x = -(\tan x – \sin 2x)\) \(= -f(x)\) Do đó \(y = \tan x – \sin 2x\) là hàm số lẻ.

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