Answer:
the range of the function (f(x) = \\sin(x)^2) is (\[0, \\infty)), and the natural domain is (\\mathbb{R}) (all real numbers)
Explanation:
The function (f(x) = \\sin(x)^2) represents the square of the sine of (x).
To find the range and natural domain of the function:
1. Natural Domain:
The natural domain of the function (f(x) = \\sin(x)^2) is all real numbers since the sine function is defined for all real values of (x).
Natural Domain: (\\mathbb{R}) (all real numbers)
2. Range:
To determine the range, we need to examine the possible values that (\\sin(x)^2) can take. Since the square of any real number is always nonnegative (greater than or equal to zero), the range of (f(x) = \\sin(x)^2) is the set of nonnegative real numbers.
Range: (\[0, \\infty)) (all nonnegative real numbers)
Therefore, the range of the function (f(x) = \\sin(x)^2) is (\[0, \\infty)), and the natural domain is (\\mathbb{R}) (all real numbers).