A range contains exactly the output values produced by inputs in the stated domain. An unrestricted vertex value belongs only if an allowed input produces it. On either side of the vertex a parabola is monotonic. On a finite interval, only an included vertex and endpoint values can determine extreme or limiting bounds. Complete the square; locate the vertex relative to the interval; evaluate both endpoint positions and any allowed vertex; decide whether each bound is actually attained. Use a filled endpoint for an allowed input and an open endpoint for an excluded one. Example For y=(x-1)2, 1<x 3, the vertex is excluded. Values increase from a limiting 0 to the attained 4, so the range is (0,4]. There is no minimum. Exam check: an open input endpoint does not automatically create an open range bound; another allowed input may produce the same output.

A range contains exactly the output values produced by inputs in the stated domain. An unrestricted vertex value belongs only if an allowed input produces it.

On either side of the vertex a parabola is monotonic. On a finite interval, only an included vertex and endpoint values can determine extreme or limiting bounds.

Complete the square; locate the vertex relative to the interval; evaluate both endpoint positions and any allowed vertex; decide whether each bound is actually attained. Use a filled endpoint for an allowed input and an open endpoint for an excluded one.

For \(y=(x-1)^2\), \(1<x\le3\), the vertex is excluded. Values increase from a limiting \(0\) to the attained \(4\), so the range is \((0,4]\). There is no minimum.

Exam check: an open input endpoint does not automatically create an open range bound; another allowed input may produce the same output.

Attained and limiting bounds

Attained and limiting bounds