Let the sets A and B denote the domain and range respectively of the function [f(x)=1 x-x]…

Question

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$$, where $$\lceil x\rceil$$ denotes the smallest integer greater than or equal to $x$. Then among the statements
(S1) :...

Accepted Answer

$f(x)=\frac{1}{\sqrt{\lceil xceil -x}}$

If $x\in I,\lceil xceil =[x]$ (greatest integer function)

If $x
otin I,\lceil xceil =[x]+1$

$\begin{gathered} \Rightarrow f(x)=\begin{gathered} \frac{1}{\sqrt{[x]-x}},x\in I \ \frac{1}{\sqrt{[x]+1-x}},x
otin I \ \Rightarrow f(x)=\frac{1}{\sqrt{-{x}}},x\in I, \ \frac{1}{\sqrt{1-{x}}},x
otin I \end{gathered} \ \Rightarrow f(x)=R-I \end{gathered}$

$\begin{gathered} f(x)=\frac{1}{\sqrt{1-{x}}},x
otin I \ \Rightarrow 01 \ \Rightarrow =(1,\infty) \ \Rightarrow A=R-I \end{gathered}$

$\begin{gathered} B=(1,\infty) \ A\cap B=(1,\infty)-N \ A\cup B
e (1,\infty) \ \Rightarrow S1 \end{gathered}$

Solutions · 1