Find the domain and the range of the real function, `f(x)=(1)/(sqrt(x+ – InterviewSolution

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1.	Find the domain and the range of the real function, `f(x)=(1)/(sqrt(x+[x]))`.
Answer» We have, `f(x)=(1)/(sqrt(x+[x]))` We know that `{{:(x+[x]gt0"for all "xgt0),(x+[x]=0",when "x=0),(x+[x]lt0"for all "xlt0):}` `:.f(x)=(1)/(sqrt(x+[x]))` is defined only when `x+[x]gt0` and this happens only when `xgt0`. `:."dom "(f)=(0,oo)`. Let y=f(x). Then, `y=(1)/(sqrt(x+[x]))impliessqrt(x+[x])=(1)/(y)." "....(i)` Now, `xgt0impliesx+[x]gt0impliessqrtx+[x]gt0implies(1)/(y)gt0impliesygt0`. `:."range "(f)=(0,oo)`. Hence, dom `(f)=(0,oo)"andrange "(f)=(0,oo)`.

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