Questions: Consider the functions f(x) = sin(2x) g(x) = sin(4x). Which statement best describes the relationship between the graphs of f(x) and g(x)? 1. The period of f is smaller than the period of g. 2. The period of f is larger than the period of g. 3. The periods of f and g are the same. 4. None of these.

Solution Steps

To determine the relationship between the periods of the functions \( f(x) = \sin(2x) \) and \( g(x) = \sin(4x) \), we need to calculate the period of each function. The period of a sine function \( \sin(kx) \) is given by \( \frac{2\pi}{k} \). We will calculate the periods of \( f(x) \) and \( g(x) \) and compare them.

1. Calculate the period of \( f(x) = \sin(2x) \).
2. Calculate the period of \( g(x) = \sin(4x) \).
3. Compare the two periods to determine the relationship.

The function \( f(x) = \sin(2x) \) has a period given by the formula:

\[ \text{Period of } f = \frac{2\pi}{k_f} = \frac{2\pi}{2} = \pi \approx 3.1416 \]

The function \( g(x) = \sin(4x) \) has a period given by the formula:

\[ \text{Period of } g = \frac{2\pi}{k_g} = \frac{2\pi}{4} = \frac{\pi}{2} \approx 1.5708 \]

Now we compare the periods of \( f \) and \( g \):

\[ \text{Period of } f \approx 3.1416 \quad \text{and} \quad \text{Period of } g \approx 1.5708 \]

Since \( 3.1416 > 1.5708 \), we conclude that the period of \( f \) is larger than the period of \( g \).

Final Answer