Question

What is the domain and range for `y=-2sqrt(9-3x) +1`?

Answer 1

The domain is `(-oo;3)` and the range is `(-oo;+1>`

The domain is the subset of `RR` for which the function value can be calculated.
In this function the only restriction for the domain is that `9-3x>=0`, because you cannot take square root of negative numbers (they are not real). After solving the inequality you get the domain `(-oo;3)`

To calculate the range you have to look at the function. There are such things in it:

1. square root of a linear function
2. multiplying by `-2`
3. adding one to the result

The first mentioned function has a range of `<0;+oo)`
The action in 2) changes the sign of the result, so the range changes to `(-oo;0>`
The last action moves the range 1 unit up, so the upper boundary changes from `0` to `1`