Certainly! Let’s determine the range of the function [tex]\( f(x) = \cos(x) \)[/tex].
1. Understand the Cosine Function:
- The function [tex]\( \cos(x) \)[/tex] is a periodic trigonometric function that outputs values based on the angle [tex]\( x \)[/tex], which can be in radians or degrees.
- It repeats its values at regular intervals of [tex]\( 2\pi \)[/tex] radians (or 360 degrees). This periodicity means it will continuously loop through its range of values.
2. Characteristics of the Cosine Function:
- The cosine function oscillates between its maximum and minimum values at [tex]\( x = 0 \)[/tex], [tex]\( x = \pi/2 \)[/tex], [tex]\( x = \pi \)[/tex], etc.
- The maximum value of [tex]\( \cos(x) \)[/tex] is [tex]\( 1 \)[/tex]: [tex]\( \cos(0) = 1 \)[/tex].
- The minimum value of [tex]\( \cos(x) \)[/tex] is [tex]\( -1 \)[/tex]: [tex]\( \cos(\pi) = -1 \)[/tex].
3. Identifying the Range:
- Therefore, the function [tex]\( \cos(x) \)[/tex] takes on all the values from [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex] inclusive.
To conclude, the range of the function [tex]\( f(x) = \cos(x) \)[/tex] is:
[tex]\[ \text{Range of } \cos(x) \text{ is } [-1, 1] \][/tex]
So, the values that [tex]\( f(x) = \cos(x) \)[/tex] can output lie within the interval:
[tex]\[ (-1, 1) \][/tex]
