Question

How do you find the domain and range of `f(x) = -sqrt(x+3)`?

Answer 1

Domain: `[-3, + oo)`
Range: `(-oo, 0]`

Explanation:

The domain of the function will be determined by the fact that, for rational numbers, you cannot take the square root of a negative number.

In other words, the expression that's under the square root must always be positive for real numbers. This means that

`x+3 >=0 implies x>=-3`

The domain of the function will thus be any values of `x in [-3, +oo)`.

The range of the function will be affected by the fact that the square root of any positive real number is always positive.

The minimum value the function can have will depend on the minimum value that `x` can take

`f(-3) = -(sqrt(-3) + 3) = -sqrt(0) = 0`

For any other value of `x`, `f(x)<0`. The range of the function will thus be `(-oo, 0]`.

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