Question

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

Answer 1

The domain is `{x \in \mathbb{R} : x\geq -3}`

The range is `(-infty, 0]`

Explanation:

The domain is `{x \in \mathbb{R} : x\geq -3}`

In fact, an even root exists if and only if its content is non negative. So, we must impose

`x+3 \geq 0 \iff x \geq -3`

As for the range, we can observe a some easy things:

• A square root is always positive, when it exists
• A square root is zero only when its content is zero. In our case, this happens only when `x = -3`
• A square root grows infinitely as its content grows infinitely

So, we know that `sqrt{x+3}` starts at zero for `x=-3`, and then grows indefinitely, since `x+3` grows indefinitely. Its range is thus `[0,\infty)`. But since we have a minus sign in front of the square root, we must reflect the range, obtaining `(-infty, 0]`