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Find whether the given function is even ...

Find whether the given function is even or odd: `f(x)=(x(sinx+tanx)/([x/pi]-1/2);` whether `[]` denotes the greatest integer function.

Text Solution

Verified by Experts

We have
`f(x)=(x(sinx+tanx))/([(x+pi)/(pi)]-(1)/(2))=(x(sinx+tanx))/([(x)/(pi)]+1-(1)/(2))=(x(sinx+tanx))/([(x)/(pi)]+0.5)`
` implies f(-x)=(-x(sin(-x)+tan(-x)))/([-(x)/(pi)]+0.5)`
`={((x(sinx+tanx))/(-1-[(x)/(pi)]+0.5)",",x ne n pi","n inZ),(0",",x=n pi",," n in Z):}`
`={(-(x(sinx+tanx))/([(x)/(pi)]+0.5)",",x ne n pi),(0",",x=n pi):}`
Thus `f(-x)= -f(x)`
Hence , f(x) is an odd function.
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