Let's analyze the function [tex]\(y = \sqrt{x - 2}\)[/tex] with the condition [tex]\(y \geq 0\)[/tex].
### Step 1: Understanding the Function
The given function is [tex]\(y = \sqrt{x - 2}\)[/tex].
- The square root function [tex]\(\sqrt{x - 2}\)[/tex] implies that the term inside the square root, [tex]\(x - 2\)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers.
### Step 2: Determining the Domain
To find the domain of the function, we need to ensure that [tex]\(x - 2 \geq 0\)[/tex].
- Solve the inequality [tex]\(x - 2 \geq 0\)[/tex]:
[tex]\[
x - 2 \geq 0 \implies x \geq 2
\][/tex]
- Thus, the domain of the function is all [tex]\(x\)[/tex] such that [tex]\(x \geq 2\)[/tex].
In interval notation, the domain is represented as:
[tex]\[
[2, \infty)
\][/tex]
### Step 3: Determining the Range
Since we are given [tex]\(y = \sqrt{x - 2}\)[/tex] and we know that the square root function outputs non-negative values, the range of [tex]\(y\)[/tex] is:
- As [tex]\(x\)[/tex] increases from 2 to infinity, [tex]\(\sqrt{x - 2}\)[/tex] increases from 0 to infinity.
- Therefore, the range of [tex]\(y\)[/tex] is all non-negative real numbers [tex]\(y \geq 0\)[/tex].
In interval notation, the range is:
[tex]\[
[0, \infty)
\][/tex]
### Final Answer
Thus, the function [tex]\(y = \sqrt{x - 2}\)[/tex] has:
- The domain [tex]\([2, \infty)\)[/tex]
- The range [tex]\([0, \infty)\)[/tex]
So, in summary:
[tex]\[
y = \sqrt{x - 2}, \quad \text{Domain: } [2, \infty), \quad \text{Range: } [0, \infty)
\][/tex]
