Problem

Analyze \(y~=~-\sqrt{x~+~1}~+~3\). Identify the root family and radicand condition, state the domain and range, locate the anchor, and give the intervals of increase and decrease.

Prompt graph of y=−√(x+1)+3 without domain, range, or monotonicity conclusions.

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Root: cube root with unrestricted radicand. Domain: (−∞,∞). Range: (−∞,∞). Anchor: central point (−1,3). Monotonicity: decreasing on all real numbers.

Root: square root with x+1≥0. Domain: [−1,∞). Range: (−∞,3]. Anchor: left endpoint (−1,3). Monotonicity: decreasing on [−1,∞); increasing nowhere.

Root: square root with x+1≤0. Domain: (−∞,−1]. Range: (−∞,3]. Anchor: right endpoint (−1,3). Monotonicity: increasing on (−∞,−1]; decreasing nowhere.

Root: square root with x+1≥0. Domain: [−1,∞). Range: [3,∞). Anchor: left endpoint (−1,3). Monotonicity: increasing on [−1,∞); decreasing nowhere.

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Analyze \(y~=~-\sqrt{x~+~1}~+~3\). Identify the root family and radicand condition, state the domain and range, locate the anchor, and give the intervals of increase and decrease.

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