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If f(x)=(2x(sinx+tanx))/(2[(x+2pi)/(pi)]...

If `f(x)=(2x(sinx+tanx))/(2[(x+2pi)/(pi)]-3)` then it is (where [.] denotes the greatest integer function)

A

odd

B

Even

C

many one and onto

D

one-one

Text Solution

Verified by Experts

The correct Answer is:
A, C
Given `f(x)=(2x(sinx+tanx))/(2[(x+2pi)/(pi)]-3) = (2x(sinx+tanx))/(2[(x)/(pi)]+1)= (x(sinx+tanx))/([(x)/(pi)] +1/2)`
`f(-x)= (x(sinx+tanx))/([-(x)/(pi)] +1/2)`
Case-I: If `x = n pi, n in I` then `[-(x/pi)]=-[x/pi]` and numerator of `f(–x)` is zero.
`implies f(–x) = 0`
Case-II: If `x pi ne n pi, n in I`
`implies f(-x)=(x(sin x+tan x)/(-[x/pi]-1+1/2)=-f(x)`
Hence `f(x)` is an odd function.
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