Problem

For \(y~=~\sqrt{x~-~2}\), state the domain, list every connected interval of increase and decrease, explicitly identify any empty category, and explain the boundary notation at \(x~=~2\).

Prompt graph of y=√(x−2) with endpoint visible and interval conclusions withheld.

Big Picture

What this problem is really about

Monotonicity must be reported on connected pieces of the actual domain, so find the square-root boundary before reading the curve from left to right. A horizontal shift moves the parent graph without reversing it. Let endpoint inclusion determine the bracket, and explicitly state when one behavior category has no interval.

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Domain: (−∞,2]. Connected rows: decreasing on (−∞,2]. Empty category: no increasing interval. Boundary: x=2 is an included right endpoint at (2,0).

Domain: (−∞,∞). Connected rows: increasing on (−∞,∞). Empty category: no decreasing interval. Boundary: x=2 is only a central anchor, not a domain boundary.

Domain: (2,∞). Connected rows: increasing on (2,∞). Empty category: no decreasing interval. Boundary: x=2 is an excluded vertical asymptote, so the interval begins with a parenthesis.

Domain: [2,∞). Connected rows: increasing on [2,∞). Empty category: no decreasing interval. Boundary: x=2 is an included endpoint at (2,0), so the monotonic interval begins with a bracket.

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For \(y~=~\sqrt{x~-~2}\), state the domain, list every connected interval of increase and decrease, explicitly identify any empty category, and explain the boundary notation at \(x~=~2\).

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