Problem

For \(y~=~1/(x~+~3)\), state the domain, list every connected interval of increase and decrease separately, explicitly identify any empty category, and explain the boundary at \(x~=~-3\).

Prompt rational graph with two branches separated at x equals negative three and interval conclusions withheld.

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Domain: (−∞,−3)∪(−3,∞). Connected rows: increasing on (−∞,−3) and increasing on (−3,∞). Empty category: no decreasing interval. Boundary: x=−3 is an excluded vertical asymptote.

Domain: (−∞,∞). Connected rows: decreasing on all real numbers. Empty category: no increasing interval. Boundary: x=−3 is an included point where the branches join.

Domain: (−∞,−3)∪(−3,∞). Connected rows: decreasing on (−∞,−3) and decreasing on (−3,∞). Empty category: no increasing interval. Boundary: x=−3 is an excluded vertical asymptote.

Domain: (−∞,−3]∪[−3,∞). Connected rows: decreasing on both intervals. Empty category: no increasing interval. Boundary: x=−3 is an included endpoint shared by both rows.

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For \(y~=~1/(x~+~3)\), state the domain, list every connected interval of increase and decrease separately, explicitly identify any empty category, and explain the boundary at \(x~=~-3\).

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