To determine the range of the square root function [tex]\(D(x) = \sqrt{x}\)[/tex], we need to identify the set of all possible values that [tex]\(D(x)\)[/tex] can take as [tex]\(x\)[/tex] varies over its domain.
Let's go through this step-by-step:
1. Identify the Domain:
The domain of the square root function [tex]\(D(x) = \sqrt{x}\)[/tex] consists of all non-negative real numbers, since the square root of a negative number is not defined within the set of real numbers. Thus, the domain of [tex]\(D(x)\)[/tex] is [tex]\(x \geq 0\)[/tex].
2. Evaluate the Function for Different Values in the Domain:
- For [tex]\(x = 0\)[/tex], [tex]\(D(0) = \sqrt{0} = 0\)[/tex].
- For [tex]\(x = 1\)[/tex], [tex]\(D(1) = \sqrt{1} = 1\)[/tex].
- For [tex]\(x = 4\)[/tex], [tex]\(D(4) = \sqrt{4} = 2\)[/tex].
3. Assess the Function's Behavior:
As [tex]\(x\)[/tex] increases, [tex]\(D(x) = \sqrt{x}\)[/tex] also increases. More specifically, as [tex]\(x\)[/tex] approaches infinity, [tex]\(\sqrt{x}\)[/tex] also grows without bound. Moreover, since [tex]\(\sqrt{x}\)[/tex] is defined to be non-negative, it can never take any negative values. Therefore, [tex]\(\sqrt{x}\)[/tex] ranges from 0 to positive infinity.
4. Conclusion about the Range:
The lowest possible value of [tex]\(D(x)\)[/tex] is 0, which occurs when [tex]\(x = 0\)[/tex].
The function has no upper bound as it can take on arbitrarily large values for sufficiently large values of [tex]\(x\)[/tex].
Thus, the range of the square root function [tex]\(D(x) = \sqrt{x}\)[/tex] is all non-negative real numbers. We represent this range mathematically as [tex]\([0, \infty)\)[/tex].
So, the range of the square root function [tex]\(D(x) = \sqrt{x}\)[/tex] is:
[tex]\[
(0, \infty)
\][/tex]
