A domain is the set of all possible input values for a function. If a function involves a square root, the domain must be restricted to ensure that the input values do not result in a negative number under the radical sign, since the square root of a negative number is not a real number.
For example, consider the function f(x) = sqrt(x). The domain of this function is all non-negative real numbers, since the square root of a negative number is not a real number. Therefore, the domain of f(x) is the set of all x values such that x is greater than or equal to zero.
Similarly, consider the function g(x) = sqrt(4 - x^2). The domain of this function is restricted to the set of all x values such that 4 - x^2 is greater than or equal to zero, since the square root of a negative number is not a real number. Solving 4 - x^2 >= 0, we get x <= 2 and x >= -2. Therefore, the domain of g(x) is the set of all x values such that -2 <= x <= 2.
In general, whenever a square root is involved in a function, the domain must be carefully considered and may need to be restricted to ensure that the function is well-defined and meaningful for all possible input values.
