Bot GPT 4-o mini To determine the period of the function \( f(x) = 2 \sin(3x + \frac{\pi}{3}) + 5 \), we focus on the sine function within the expression.
The general form of a sine function is \( \sin(kx) \), where \( k \) is a constant that affects the period of the function. The period \( P \) of the sine function is given by the formula:
\[ P = \frac{2\pi}{|k|} \]
In our function, we have:
\[ f(x) = 2 \sin(3x + \frac{\pi}{3}) + 5 \]
Here, the term \( k \) corresponds to the coefficient of \( x \) in the argument of the sine function, which is \( 3 \).
Calculating the period:
\[ P = \frac{2\pi}{|3|} = \frac{2\pi}{3} \]
Therefore, the period of the function \( f(x) \) is
\[ \frac{2\pi}{3}. \]
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