Problem

Analyze \(y~=~\sqrt{x~-~2}~+~5\). 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−2)+5 without domain, range, or monotonicity conclusions.

Big Picture

What this problem is really about

Start with the root family, because its parity controls whether there is an endpoint at all. For the square-root form, solve the radicand condition, evaluate the boundary to locate the anchor, and use the outside shift to move the range. The parent curve's direction is unchanged by horizontal and vertical translations, so monotonicity follows from that same structure.

Answer choices

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Root: cube root with unrestricted radicand. Domain: (−∞,∞). Range: (−∞,∞). Anchor: central point (2,5). Monotonicity: increasing on all real numbers.

Root: square root with x−2≤0. Domain: (−∞,2]. Range: [5,∞). Anchor: right endpoint (2,5). Monotonicity: decreasing on (−∞,2]; increasing nowhere.

Root: square root with x−2≥0. Domain: [2,∞). Range: [5,∞). Anchor: left endpoint (2,5). Monotonicity: increasing on [2,∞); decreasing nowhere.

Root: square root with x−2≥0. Domain: [2,∞). Range: (−∞,5]. Anchor: left endpoint (2,5). Monotonicity: decreasing on [2,∞); increasing nowhere.

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Analyze \(y~=~\sqrt{x~-~2}~+~5\). 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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