Find the Domain and the Range of the function y = \sqrt{1-x^2}


For real values of {y}, the quantity {1-x^2} cannot be negative.


Subtract {1} from both sides of this inequality.


Multiply this inequality with {-1}


(Note that multiplying by a negative number reverses the inequality.)

Since both sides of this inequality are non-negative, we can take the principal square root. Doing so, we get


\sqrt{x^2} is another definition of |x|

So, we can write this inequality as |x|\leq{1}


Therefore, {x} is greater than or equal to {-1} and less than or equal to {1}

Hence, Domain =[-1,1]

Graphical representation of the Domain of sqrt(1-x^2)

Now let’s find the range of this function.

As {x} varies from {-1} to {1}, {x^2} varies from {0} to {1}

Therefore, {0}\leq{x^2}\leq{1}

Multiply this inequality with -1


Now add {1} to this inequality.


The quantity {1-x^2} is non-negative on the domain of this function. So, taking the principal square root, we get


Therefore, Range =[0,1]

Hence, the Domain of the function \boldsymbol{{y}=\sqrt{1-x^2}} is \boldsymbol{[-1,1]} and its Range is \boldsymbol{[0,1]}.


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Domain and Range of function y = sqrt(1-x^2)
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