Find the domain of the real-valued function f | vedclass

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Question

(B) For the function $f(x) = ([x]^2 - [x] - 2)^{-1/2}$ to be defined,the expression inside the square root must be strictly positive: $[x]^2 - [x] - 2

Vedclass · Accepted Answer

$R - [-1, 3)$

Vedclass · Answer

(B) For the function $f(x) = ([x]^2 - [x] - 2)^{-1/2}$ to be defined,the expression inside the square root must be strictly positive: $[x]^2 - [x] - 2 > 0$ Factorizing the quadratic expression: $([x] - 2)([x] + 1) > 0$ This inequality holds when $[x]  2$. Case $1$: If $[x] Case $2$: If $[x] > 2$,then $[x] \geq 3$,which implies $x \geq 3$. Combining these,the domain is $(-\infty, -1) \cup [3, \infty)$. This can be written as $R - [-1, 3)$.

Find the domain of the real-valued function $f(x) = ([x]^2 - [x] - 2)^{-1/2}$,where $[\cdot]$ denotes the greatest integer function.

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