Question

How do you find the domain and range of $\arcsin \left( {1 - {x^2}} \right)$?

Answer 1

Hint: We will first find the general range of inverse of sin function and then we will just apply the domain part to the inside function given and then find the required values of x.

Complete step-by-step answer:
We are given that we are required to find the domain and range of $\arcsin \left( {1 - {x^2}} \right)$.
Let us write this as: \[y = si{n^{ - 1}}\left( {1 - {x^2}} \right)\]
If we take the sine function on both the sides, we will then obtain the following expression:-
\[ \Rightarrow \sin y = 1 - {x^2}\]
Now, we know that sine function takes the value between -1 and 1.
Therefore, both the left hand and right hand side of both sides in the above mentioned expression will lie between – 1 and 1.
$ \Rightarrow - 1 \leqslant 1 - {x^2} \leqslant 1$
Multiplying the whole equation by negative, we will then obtain the following expression:-
$ \Rightarrow - 1 \leqslant {x^2} - 1 \leqslant 1$
Adding 1 in the whole equation above, we will then obtain the following equation:-
$ \Rightarrow 1 - 1 \leqslant {x^2} - 1 + 1 \leqslant 1 + 1$
On simplifying the values, we will then obtain the following equation:-
$ \Rightarrow 0 \leqslant {x^2} \leqslant 2$
Taking square – root in whole equation in above expression, we will then obtain the following equation:-
$ \Rightarrow 0 \leqslant x \leqslant \sqrt 2 $
Therefore, we have the domain of the given function $\arcsin \left( {1 - {x^2}} \right)$ as $[0,\sqrt 2 ]$.
Now, let us discuss the range.
We know that the principal value of the inverse of sine function always lies between $\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$.

Therefore, the range of given function $\arcsin \left( {1 - {x^2}} \right)$ is $\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$.

Note:
The students must know the definitions of both the range and the domain of a function.
Domain of a function is the set of those values which can be put in the given function without creating any problems. For example: In the given function the possible values of x is domain.
Range of a function is the set of those values which can be received as the image. For example:- In the given function, the possible values of y is the set of range.