To determine which of the given trigonometric functions has an amplitude of 2, we need to recall the concept of amplitude for sine and cosine functions. The amplitude of a function of the form [tex]\( f(x) = a \sin(x) \)[/tex] or [tex]\( f(x) = a \cos(x) \)[/tex] is the absolute value of the coefficient [tex]\( a \)[/tex], i.e., [tex]\( |a| \)[/tex].
Here are the given functions with an analysis of their amplitudes:
Option F: [tex]\( f(x) = 2 \sin(x) \)[/tex]
- The coefficient of [tex]\( \sin(x) \)[/tex] is 2.
- Therefore, the amplitude is [tex]\( |2| = 2 \)[/tex].
Option G: [tex]\( f(x) = 2 \tan(x) \)[/tex]
- The tangent function [tex]\( \tan(x) \)[/tex] does not have an amplitude because it can take on all real values between [tex]\( -\infty \)[/tex] and [tex]\( \infty \)[/tex] over its period.
- Thus, this option does not have a defined amplitude.
Option H: [tex]\( f(x) = \sin \left(\frac{1}{2} x\right) \)[/tex]
- The coefficient of [tex]\( \sin(x) \)[/tex] here is 1.
- Therefore, the amplitude is [tex]\( |1| = 1 \)[/tex].
Option J: [tex]\( f(x) = \cos(2x) \)[/tex]
- The coefficient of [tex]\( \cos(x) \)[/tex] here is 1.
- Therefore, the amplitude is [tex]\( |1| = 1 \)[/tex].
Option K: [tex]\( f(x) = \frac{1}{2} \cos(x) \)[/tex]
- The coefficient of [tex]\( \cos(x) \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
- Therefore, the amplitude is [tex]\( \left|\frac{1}{2}\right| = 0.5 \)[/tex].
After analyzing all these options, we find that the only function with an amplitude of 2 is:
Option F: [tex]\( f(x) = 2 \sin(x) \)[/tex].
Therefore, the trigonometric function that has an amplitude of 2 is:
[tex]\[ \boxed{f(x) = 2 \sin(x)} \][/tex]
