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where f(x)= [1/ sin{x} ] and [.]=greatest integer and {}=fractional part function? explain and solve.
Hi Shelley,
We know 0≤{x}<1
But as {x} is in the Denominator in this problem, we have 0<{x}<1
So sin0<sin{x}<sin1
or 0<sin{x}<sin1.
Also sin1>sin(pi/6)
or sin1>0.5.
Hence 0<sin{x}< (a number that is more than 0.5)
Hence 1/sin{x} can take all values more than 1/sin1
1/sin1 = 1.xyz....
Hence [1/sin{x}] can take all positive integral values.
And Hence the Range of this function is the set of all Natural Numbers.
Regards,
Ashwin (IIT Madras).
fractional part of x means x is always between 0 and 1 radian meaning always between 0 and 57.2 degrees so as x increases from 0 to 57.2 sin x increases and 1/ sin x decreases so the integral part of it also decreases ...so its a decreasing function,as the final function is in integral part , the range will be only integers , now when x = 0 f (x) becomes infinite and when x keeps thaon increasin from 0 to 57.2 degrees the integral values keep decreasin form infinite to ... [1/sin(57.2) degrees] thats integral part of 1.19 = 1 , so the range is from infinite to 1 all the integers between them,meaning the range is natural number..... APPROVE IF YOU HAVE UNDERSTOOD THE SOLUTION
Regards
Noaman Ali
IIT-Roorkee
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