If [cdot] denotes the greatest integer functi | vedclass

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Question

(A) The function is given by $f(x) = \frac{\sin([x]\pi) + 	an([x]\pi)}{1 + [x]^2 + [x]^4}$.Since $[x]$ is the greatest integer function,it is defined

Vedclass · Accepted Answer

$R, \{0\}$

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(A) The function is given by $f(x) = \frac{\sin([x]\pi) + 	an([x]\pi)}{1 + [x]^2 + [x]^4}$.Since $[x]$ is the greatest integer function,it is defined for all real numbers $x \in R$.Thus,the domain of the function is $R$.For any integer $n$,$[x] = n$,where $n \in Z$.Substituting this into the function,we get $f(x) = \frac{\sin(n\pi) + 	an(n\pi)}{1 + n^2 + n^4}$.We know that for any integer $n$,$\sin(n\pi) = 0$ and $	an(n\pi) = 0$.Therefore,$f(x) = \frac{0 + 0}{1 + n^2 + n^4} = 0$ for all $x \in R$.Since the function value is always $0$,the range of the function is the singleton set $\{0\}$.Hence,the domain is $R$ and the range is $\{0\}$.

If $[\cdot]$ denotes the greatest integer function,then the domain and range of the function $f(x) = \frac{\sin([x]\pi) + \tan([x]\pi)}{1 + [x]^2 + [x]^4}$ are respectively

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