To determine the domain of the function [tex]\( f(x) = \sqrt{121 - x^2} \)[/tex], we need to ensure that the expression under the square root, [tex]\( 121 - x^2 \)[/tex], is non-negative.
This is because the square root function is only defined for non-negative arguments. Therefore, we need to solve the inequality:
[tex]\[
121 - x^2 \geq 0
\][/tex]
First, we rearrange the inequality to isolate [tex]\( x^2 \)[/tex]:
[tex]\[
121 \geq x^2
\][/tex]
Next, we recognize that [tex]\( 121 \)[/tex] is a perfect square and can be written as [tex]\( 11^2 \)[/tex]. So our inequality becomes:
[tex]\[
11^2 \geq x^2
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
\sqrt{11^2} \geq \sqrt{x^2}
\][/tex]
This simplifies to:
[tex]\[
11 \geq |x|
\][/tex]
The absolute value inequality [tex]\( 11 \geq |x| \)[/tex] can be split into two separate inequalities:
[tex]\[
-11 \leq x \leq 11
\][/tex]
Thus, the domain of the function [tex]\( f(x) = \sqrt{121 - x^2} \)[/tex] includes all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is between [tex]\(-11\)[/tex] and [tex]\(11\)[/tex], inclusive.
Therefore, the domain of the function in interval notation is:
[tex]\[
[-11, 11]
\][/tex]
This means the domain is [tex]\( \boxed{[-11, 11]} \)[/tex].
