To determine the range of the function [tex]\( f(x) = \sin(x) \)[/tex], we need to consider the behavior and output values of the sine function for all real numbers [tex]\( x \)[/tex].
The sine function, [tex]\( \sin(x) \)[/tex], is a periodic function that oscillates between certain values. Specifically, the function takes on values that form a wave pattern, repeating every [tex]\( 2\pi \)[/tex] radians.
Here are the key observations to make:
1. Amplitude: The maximum and minimum values of [tex]\( \sin(x) \)[/tex] are 1 and -1, respectively. This means for any input [tex]\( x \)[/tex], [tex]\( \sin(x) \)[/tex] will always generate output values that lie within this range.
2. Output Values: For [tex]\( \sin(x) \)[/tex], the output values range from -1 to 1 inclusive. It cannot produce values outside this interval.
3. Periodicity: The sine function repeats its values every [tex]\( 2\pi \)[/tex] radians, but this periodicity does not affect the range, only how often the values repeat.
So, the range of the sine function is the set of all [tex]\( y \)[/tex] values such that [tex]\( -1 \leq y \leq 1 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\text{The set of all real numbers } -1 \leq y \leq 1
\][/tex]
