Answer:

[tex]\boxed{\begin{aligned}& \qquad \:\sf \: Domain\:of\:f(x): x \: \in \: R \: \qquad \: \\ \\& \qquad \:\sf \: Range\:of\:\dfrac{1}{2 - sinx} \: \in \: \left[\dfrac{1}{3} \: , \: 1\right] \end{aligned}}  \\ [/tex]

Step-by-step explanation:

Given function is

[tex]\sf \: f(x) = \dfrac{1}{2 - sinx} \\ [/tex]

We know,

Domain of a function f(x) is defined as set of those real values of x for which function is well defined.

As we know,

[tex]\sf \: - 1 \leqslant sinx \leqslant 1 \\ [/tex]

[tex]\implies \sf \:2 - sinx \: \ne \: 0 \\ [/tex]

[tex]\implies\boxed{\sf\: \bf \:Domain\:of\:f(x): x \: \in \: R \: \: } \\ [/tex]

Range

We know,

[tex]\sf \: - 1 \leqslant sinx \leqslant 1 \\ [/tex]

[tex]\sf \: - 1 \leqslant - sinx \leqslant 1 \\ [/tex]

[tex]\sf \: 2- 1 \leqslant 2 - sinx \leqslant 2 + 1 \\ [/tex]

[tex]\sf \: 1 \leqslant 2 - sinx \leqslant 3 \\ [/tex]

[tex]\implies \sf \:\dfrac{1}{3} \leqslant \dfrac{1}{2 - sinx} \leqslant 1 \\ [/tex]

[tex]\implies \sf \: \dfrac{1}{2 - sinx} \: \in \: \left[\dfrac{1}{3} \: , \: 1\right] \\ [/tex]

[tex]\implies \sf \: \boxed{\bf\:Range\:of\:\dfrac{1}{2 - sinx} \: \in \: \left[\dfrac{1}{3} \: , \: 1\right] \: } \\ [/tex]

Hence,

[tex]\implies \sf \:\boxed{\begin{aligned}& \qquad \:\sf \: Domain\:of\:f(x): x \: \in \: R \: \qquad \: \\ \\& \qquad \:\sf \: Range\:of\:\dfrac{1}{2 - sinx} \: \in \: \left[\dfrac{1}{3} \: , \: 1\right] \end{aligned}}  \\ [/tex]