Problem

Determine the domain and range of \(y=\sqrt{x-4}+2\).

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What this problem is really about

For an even-root function, the inside expression controls which inputs are real while the outside shift controls the output boundary. We’ll solve the nonnegative-radicand condition, evaluate the boundary input to locate the included endpoint, and transform the parent output bound into the range.

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family=square-root; input condition=x−4≥0; solved condition=x≥4; endpoint=(4,2); output direction=y≥2; domain=[4,∞); range=[2,∞)

family=square-root; input condition=x−4≥0; solved condition=x≥4; endpoint=(4,2); output direction=y≥0 after vertical shift is ignored; domain=[4,∞); range=[0,∞)

family=square-root; input condition=x≥2 from vertical shift; solved condition=x≥2; endpoint=(2,2); output direction=y≥2; domain=[2,∞); range=[2,∞)

family=square-root; input condition=x−4>0; solved condition=x>4; endpoint=(4,2) excluded; output direction=y>2; domain=(4,∞); range=(2,∞)

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Determine the domain and range of \(y=\sqrt{x-4}+2\).

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