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Write the piecewise definition of the following functions. (i) f(x)= [sqrt(x)] "(ii) " f(x)=[tan^(-1)x] "(iii) " f(x)=[log_(e)x] In each case [.] denotes the greatestinteger function. |
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Answer» Solution :(i)`f(x)=[sqrt(x)], x ge 0.` `[sqrt(x)]=0 " if " sqrt(x) in[0,1) " or " x in [0,1)` `[sqrt(x)]=1 " if " sqrt(x) in[1,2) " or " x in [1,4)` `[sqrt(x)]=2 " if " sqrt(x) in[2,3) " or " x in [4,9)` and so on. ` :. f(x) ={(0"," x in[0,1)),(1"," x in[1,4)),(2"," x in[4,9)),(3"," x in[9,16)),(...),(...):}` (ii) `f(x)=[tan^(-1)x]` We KNOW that `tan^(-1)x in(-(pi)/(2),(pi)/(2))` ` :. " possible values of " [tan^(-1) x]" are " -2,-1,0,1.` If `[tan^(-1)x]= -2," then " tan^(-1)x in (-(pi)/(2),-1) " or " x in (-oo,-tan1) ` If `[tan^(-1)x]= -1," then " tan^(-1)x in [-1,0) " or " x in [-tan1,0) ` If `[tan^(-1)x]= 0," then " tan^(-1)x in [0,1) " or " x in [0,tan 1) ` If `[tan^(-1)x]= 1," then " tan^(-1)x in [1,(pi)/(2)) " or " x in [tan 1, oo) ` (III) `f(x) =[log_(e)x]` We know that `log_(e) in (-oo,oo)`. ` :. [log_(e)x] in Z," i.e., " [log_(e)x]` TAKES all integral values. If `[log_(e)x] =N,n in Z " then " log_(e)x in [n,n+1) " or " x in [e^(n),e^(n+1))`. Therefore, `[log_(e)x]={(...),(...),(-1"," x in[e^(-1),1)),(0"," x in[1,e)),(1"," x in[e,e^(2))),(2"," x in[e^(2),e^(3))),(3"," x in[e^(3),e^(4))),(...),(...):}` |
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