To determine the range of the function [tex]\( f(x) = \sin(x) \)[/tex], we first need to understand the behavior of the sine function.
1. Definition and Periodicity: The sine function is defined for all real numbers and is periodic with a period of [tex]\(2\pi\)[/tex]. This means that [tex]\( \sin(x) \)[/tex] repeats its values every [tex]\(2\pi\)[/tex].
2. Amplitude: The sine function oscillates between its maximum and minimum values. The maximum value of [tex]\( \sin(x) \)[/tex] is 1 and the minimum value is -1. This pattern repeats indefinitely.
3. Range Identification:
- Since the maximum value [tex]\( \sin(x) \)[/tex] can achieve is 1 and the minimum value is -1, we can conclude that [tex]\( \sin(x) \)[/tex] will never take a value outside the interval from -1 to 1.
- This means that regardless of what value [tex]\( x \)[/tex] takes, [tex]\( y = \sin(x) \)[/tex] will always lie within the range [-1, 1].
Therefore, the correct answer is:
[tex]\[ \text{the set of all real numbers } -1 \leq y \leq 1. \][/tex]
So the range of [tex]\( f(x) = \sin(x) \)[/tex] is:
[tex]\[ \boxed{2 \text{ corresponding to } -1 \leq y \leq 1.} \][/tex]
