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]#

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