Question

Find the domain of \[\sqrt{\left( x-4 \right)\times \sin \left( x-2 \right)}\]?

Answer 1

Hint: In the given question, we have been asked to find the domain of the given function. In order to find the domain of any given function, we will use the definition of Domain. To find the domain of any given function, we will need to find the set of all the possible values which will satisfy the given function in the question.

Complete step by step answer:
We have the given function,
\[\sqrt{\left( x-4 \right)\times \sin \left( x-2 \right)}\]
Domain is basically the set of all the values that the \[x\] in the given function can take, and those values of \[x\] will satisfy the given function.
Square root is the function here taken as a real value function, so only defined for non-negative values.
So the domain of the given function are only those values of \[x\] for which the given function is non-negative. Any polynomial function or variables without denominator, mostly the domain is set of all real numbers.
Here,
To find the domain,
We know that the given function;
The domain is such that the argument,
\[\Rightarrow x-4\ge 0\Rightarrow x\ge 4\]
And,
As we know that,
\[f\left( x \right)\]is defined at \[\sin \theta \ge 0\].
Here,
\[\theta =x-2\]
So,
\[x-2\ge 0\Rightarrow x\le 2\]
Therefore,
\[x\in R:\left\{ {} \right.n\ge 0\ and\ 2-\pi \le 2\pi n\le x\le 2\ and\ n\in Z\]
OR
\[\left\{ n\le -1\ and\ 2\le 2\pi n+x\le 2+\pi \ and\ n\in Z \right.\]
OR
\[\left\{ 4 \right.OR
\[\left\{ x=4 \right.\]

Therefore, domain of the function is \[x\in R\].

Note: While solving these types of questions where we would have to find the domain of any given function, you need to know that or you have to keep in mind that all the values that go into a given function are called the domain of function. You can also identify the domain from the relation set, which will show the collection of all possible ordered pairs for the given function. The domain is always the set of all the first element i.e. x coordinates of ordered pairs.