Find whether the given function is even or odd function, where f(x)=x(sinx+tanx)[x+ππ]−12, where x≠nπ, where [] denotes the greatest integer function.
f(x) is an odd function
f(x) is neither even nor odd function
f(x) is both even and odd function
f(x) is an even function
A
f(x) is an odd function
B
f(x) is both even and odd function
C
f(x) is neither even nor odd function
D
f(x) is an even function
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Solution
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f(x)=x(sinx+tanx)[x+ππ]−12=x(sinx+tanx)[xπ]+1−12 f(x)=x(sinx+tanx)[xπ]+0.5 ⇒f(−x)=−x(sin(−x)+tan(−x))[−xπ]+0.5 ⇒f(−x)=x(sinx+tanx)−1−[xπ]+0.5 Hence, f(−x)=−⎛⎜
⎜⎝x(sinx+tanx)[xπ]+0.5⎞⎟
⎟⎠ ⇒f(−x)=−f(x) Hence, f(x) is an odd function (if x≠nπ).
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