To determine the period of the function [tex]\( y = 1 + \tan \left( \frac{1}{2} x \right) \)[/tex], we need to analyze the periodic component within the function, which in this case is the tangent, [tex]\( \tan \left( \frac{1}{2} x \right) \)[/tex].
The general form of the tangent function is [tex]\( \tan(bx) \)[/tex], where [tex]\( b \)[/tex] is a constant that affects the period of the function. The period [tex]\( T \)[/tex] of [tex]\( \tan(bx) \)[/tex] is given by:
[tex]\[ T = \frac{\pi}{|b|} \][/tex]
In the given function [tex]\( \tan \left( \frac{1}{2} x \right) \)[/tex], we identify that [tex]\( b = \frac{1}{2} \)[/tex]. We now substitute [tex]\( b \)[/tex] into the period formula:
[tex]\[ T = \frac{\pi}{\left| \frac{1}{2} \right|} \][/tex]
[tex]\[ T = \frac{\pi}{1/2} \][/tex]
[tex]\[ T = \pi \cdot \frac{2}{1} \][/tex]
[tex]\[ T = 2\pi \][/tex]
Therefore, the period of the function [tex]\( y = 1 + \tan \left( \frac{1}{2} x \right) \)[/tex] is [tex]\( 2\pi \)[/tex].
Hence, the correct answer is:
C. [tex]\( 2\pi \)[/tex]
