Question

The domain of $f\left( x \right)=\dfrac{1}{1-2\sin x}$ is
A. $\left( -\infty ,\infty \right)$
B. $R-\left\{ n\pi +{{\left( -1 \right)}^{n}}\dfrac{\pi }{6};n\in Z \right\}$
C. $R-\left\{ n\pi ;n\in Z \right\}$
D. $R-\left\{ n\pi +{{\left( -1 \right)}^{n}}\dfrac{\pi }{3};n\in Z \right\}$

Answer 1

Hint: Here we have been given a function and we have to find the domain in which it lies. Firstly as we know for a fraction value the denominator should never be zero if we want the function to be defined so using this concept we will form a statement. Then we will simplify the statement and get the value of the variable which should not be equal to zero. Finally we will subtract that value from the real numbers and get our desired answer.

Complete step-by-step solution:
The function is given as follows:
$f\left( x \right)=\dfrac{1}{1-2\sin x}$
As we now that in order to be defined this function denominator should not be equal to $0$ so we can write,
$1-2\sin x\ne 0$
$\Rightarrow 2\sin x\ne 1$
Simplifying further we get,
$\Rightarrow \sin x\ne \dfrac{1}{2}$
Take sine function on the right,
$\Rightarrow x\ne {{\sin }^{-1}}\left( \dfrac{1}{2} \right)$
We know that $\sin \left( n\pi +{{\left( -1 \right)}^{n}}\dfrac{\pi }{6} \right)=\dfrac{1}{2}$ , $\forall n\in Z$ replacing it above we get,
$\Rightarrow x\ne {{\sin }^{-1}}\sin \left( n\pi +{{\left( -1 \right)}^{n}}\dfrac{\pi }{6} \right)$
$\Rightarrow x\ne n\pi +{{\left( -1 \right)}^{n}}\dfrac{\pi }{6},\forall n\in Z$
Therefore we get that the function variable should not be equal to above values so we will subtract the above value by the Real numbers and get,
$R-\left\{ n\pi +{{\left( -1 \right)}^{n}}\dfrac{\pi }{6},\forall n\in Z \right\}$
So the function is defined for the above value and hence it will be its domain.
Hence the correct option is (B).

Note: Function is the relations where for each input there is an output. Functions are formed between two sets such that they are non-empty sets. Where the one set is the domain and the other set is the co-domain. Condition for a function is that for every element in the domain there exists an unique element in the co-domain. Domain is the set of all possible values that qualifies as the input functions. Domains in a fraction are found by taking the values in the denominator not equal to zero and in square root the digit under the bracket should be positive.