To determine an [tex]\( x \)[/tex]-intercept for the function [tex]\( y = \cos \left(\frac{1}{2} x \right) \)[/tex], we need to find the value of [tex]\( x \)[/tex] at which the function [tex]\( \cos \left(\frac{1}{2} x \right) \)[/tex] equals zero.
The cosine function equals zero at specific angles given by:
[tex]\[ \cos \theta = 0 \text{ when } \theta = (2n+1) \frac{\pi}{2} \][/tex]
where [tex]\( n \)[/tex] is any integer.
For our function [tex]\( y = \cos \left(\frac{1}{2} x \right) \)[/tex], we set:
[tex]\[ \frac{1}{2} x = (2n+1) \frac{\pi}{2} \][/tex]
To solve for [tex]\( x \)[/tex], we multiply both sides by 2:
[tex]\[ x = (2n+1) \pi \][/tex]
Now, we assess the given choices:
A. 0
B. [tex]\( \frac{\pi}{2} \)[/tex]
C. [tex]\( \pi \)[/tex]
D. [tex]\( 2 \pi \)[/tex]
We can see that [tex]\( x = \pi \)[/tex] fits the form of [tex]\( x = (2n+1) \pi \)[/tex] where [tex]\( n = 0 \)[/tex]:
[tex]\[ x = (2 \cdot 0 + 1) \pi = \pi \][/tex]
Thus, the correct answer is:
C. [tex]\( \pi \)[/tex]
