To determine the range of the function \( f(x) = \sin(x) \), we need to understand the behavior of the sine function across all possible input values, \( x \).
1. Understanding the Sine Function:
- The sine function, \( \sin(x) \), is a periodic function, which means it repeats its values in regular intervals. Specifically, for the sine function, this interval is \( 2\pi \). This means \( \sin(x + 2k\pi) = \sin(x) \) for any integer \( k \).
2. Behavior of the Sine Function:
- The sine function oscillates smoothly between its maximum and minimum values. The highest value \( \sin(x) \) can reach is 1, and the lowest value it can reach is -1. This is because the sine of an angle in a right triangle (which is defined as the ratio of the opposite side to the hypotenuse) cannot be greater than 1 or less than -1.
3. Range of the Sine Function:
- Since \( \sin(x) \) achieves every value between -1 and 1 inclusive, and repeats these values for every \( 2\pi \) interval, the set of all possible output values, or the range, of \( \sin(x) \) is \( \{ y \in \mathbb{R} \mid -1 \leq y \leq 1 \} \).
Thus, the range of the function [tex]\( f(x) = \sin(x) \)[/tex] is [tex]\([-1, 1]\)[/tex].
