It seems you have to find the ranges of the functions given: y = 3sin(x - π), y = 1 - sin(x), y = 3+4cos(x-π), y = 2+cot(x).
1) y = 3sin(x-π).
The function sine returns a value between -1 and 1, including those limits, so the range interal is [-1,1] no matter what the input be.
Multiplying the function by the factor 3, stretches the range by that factor.
So, the range of the given function is [-3,3].
2) y = 1 - sin(x)
Againt the range of sin(x) is [-1,1], then when you subtract that from 1, the limits of 1 - sin(x) are: 1 - (-1) = 2 and 1 - 1=0, and any other value wiill be in that range.
So, the answer is [0,2]
3) y = 3 + 4cos(x - π).
The range of the cosine function is also [-1,1], so the range of cos(x - π) is [-1,1].
By multiplying by 4 you stretch the range by that factor, so the new range is [-4,4].
By adding 3, the new borders are 3 - 4 = -1 and 3 + 4 = 7, so the range of the given function is: [-1, 7].
4) y = 2+cot(x)
The range of cot(x) is (-∞,∞), so adding 2 does not modify the range
So, the range for the given function is (-∞,∞).
Be sure to understand how this work. For both functions sine and cosine the range is [-1,1], when you mutiply by a constant factor you scale the range by that factor and when you add a constant you translate the range by that number of units.