Answer:
The period of the cotangent function, cot(x), is π.
To determine which of the given functions have the same period as cot(x), we need to check if the coefficient of x inside the cotangent function affects the period.
Let's analyze each function:
1. y = 3cot(x)
The coefficient of x inside the cotangent function is 1, which means the period remains π. Therefore, this function has the same period as cot(x).
2. y = cot(3x - 1)
The coefficient of x inside the cotangent function is 3, which affects the period. The period of cot(3x - 1) is π/3. Therefore, this function does not have the same period as cot(x).
3. y = cot(x + 3π)
The term added to x inside the cotangent function does not affect the period. Therefore, this function has the same period as cot(x), which is π.
4. y = 4cot(0.5x)
The coefficient of x inside the cotangent function is 0.5, which affects the period. The period of cot(0.5x) is 2π/0.5 = 4π. Therefore, this function does not have the same period as cot(x).
5. y = 5cot(x) + 4
The coefficient of x inside the cotangent function is 1, which means the period remains π. Therefore, this function has the same period as cot(x).
To summarize, the functions that have the same period as cot(x) are:
- y = 3cot(x)
- y = cot(x + 3π)
- y = 5cot(x) + 4