Question

What is the domain and range of `r(x)= -3sqrt(x-4) +3`?

Answer 1

Domain: `[4, +oo)`
Range: `(-oo, 3]`

Explanation:

Your function is defined for any value of `x` that will not make the expression under the square root negative.

In other words, you need to have

`x-4>=0 implies x>=4`

The domain of the function will thus be `[4, +oo)`.

The expression under the square root will have a minimum value at `x = 4`, which corresponds to maximum value of the function

`r = -3 * sqrt(4-4) + 3`

`r = -3 * 0 + 3`

`r = 3`

For any value of `x>4`, you have `x-4>0` and

`r = underbrace(-3 * sqrt(x-4))_(color(blue)(<-3)) + 3 implies r < 3`

The range of the function will thus be `(-oo, 3]`.

graph{-3 * sqrt(x-4) + 3 [-10, 10, -5, 5]}