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