InterviewSolution
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InterviewSolution
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Write the function `f(x) ={sinx}` where {.} denotes the fractional part function) in piecewise definition. |
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Answer» We have `f(x)={sinx}` Clearly, `{sinx}=0` if sin x is integer. ` :. {sinx}=0 " when " x=(n pi)/(2)` Now when ` sin x in (0,1)` ` {sin x} =sin x-[sin x]=sin x -0=sinx` When ` sinx in (-1,0),` ` {sin x} =sin x-[sin x]=sin x -(-1)=sinx+1` Thus, `f(x)={(0","sinx in{-1,0,1}),(sinx","sinx in (0,1)),(sinx+1","sinx in (-1,0)):}` `f(x)={(0",",x=(n pi)/(2)","n in Z),(sinx",",x in underset(n in Z)(cup)(2n pi","(2n+1)pi)),(sinx+1",",x in underset(n in Z)(cup)"((2n+1)"pi","(2n+2)pi")"):}` |
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