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

1 Answer
KillerBunny
May 7, 2018

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]

Related questions
See all questions in Domain and Range of a Function
Impact of this question
1636 views around the world
You can reuse this answer
Creative Commons License