91Ó°ÊÓ

In the realm of mathematics, the square root function is a type of function that involves finding the number which, when multiplied by itself, gives the original number. For instance, if we consider the function \( f(x) = \sqrt{81-x^2} \), it contains a square root, meaning we are interested in the values under the square root symbol, specifically \( 81-x^2 \). When defining a square root function, it is crucial to ensure the expression under the square root is non-negative. This is because you cannot have the square root of a negative number when dealing with real numbers (more on this later).

Thus, to determine the domain of a square root function like \( f(x) = \sqrt{81-x^2} \), we must ensure that the expression inside, such as \( 81 - x^2 \), remains \( \geq 0 \). This constraint ensures we only deal with real, non-negative outputs, serving as the first rule in determining the domain of any function involving a square root.

Inequalities play a significant role in determining the domain of functions like \( f(x) = \sqrt{81-x^2} \). An inequality is a mathematical statement that relates two values, indicating that one is greater than, less than, or possibly equal to another. To find where \( 81-x^2 \) is non-negative, we translate the constraint into the inequality \( 81-x^2 \geq 0 \).

Solving this inequality involves a few steps. First, rearrange it to \( x^2 \leq 81 \). This new expression indicates we are searching for values of \( x \) that do not exceed 81 when squared. Solving \( x^2 \leq 81 \) results in finding the boundary by taking the square root of both sides, leading us to \( -9 \leq x \leq 9 \).

This result implies that \( x \) can be any real number between \(-9\) and \(9\), including \(-9\) and \(9\) themselves. Inequality solutions like these define the domain for our square root function by establishing the permissible values for \( x \).

Real numbers are the set of numbers that include all the rational and irrational numbers, essentially encompassing most numbers that you encounter in everyday life. In the context of the domain of functions, specifically our function \( f(x) = \sqrt{81-x^2} \), we focus only on values within this real number system.

The domain of a function involves identifying all possible \( x \) values that result in a real output for \( f(x) \). Since square roots of negative numbers are not defined within real numbers, the constraint \( 81-x^2 \geq 0 \) limits us to considering only real numbers for \( x \).

After solving the inequality, \( x^2 \leq 81 \), and determining that \( x \) ranges from \(-9\) to \(9\), including both endpoints, we say the domain of \( f(x) = \sqrt{81-x^2} \) is \([-9, 9]\) in the real number line. This range indicates those values of \( x \) maintain the expression under the square root as non-negative, ensuring that the output remains a real number.