The range contains only outputs actually attained by the function on its stated domain. An excluded input endpoint may still give an output attained at a different allowed input. For a quadratic, locate its vertex, check whether it belongs to the domain, and compare endpoint outputs or limiting outputs. A vertex outside the domain cannot supply an attained extreme. For f(x)=(x-2)2+1, 0<x 5, the vertex (2,1) is allowed. The upper endpoint gives 10, so the range is [1,10]; the excluded input 0 does not remove output 5, since f(4)=5. For a monotonic branch, track the endpoint outputs in order and transfer open/closed endpoints to the corresponding range boundaries. Check: on 3<x 5, the same function has range (2,10], not [1,10]. The vertex is no longer available.

The range contains only outputs actually attained by the function on its stated domain. An excluded input endpoint may still give an output attained at a different allowed input.

For a quadratic, locate its vertex, check whether it belongs to the domain, and compare endpoint outputs or limiting outputs. A vertex outside the domain cannot supply an attained extreme.

For \(\mathrm f(x)=(x-2)^2+1\), \(0<x\le5\), the vertex \((2,1)\) is allowed. The upper endpoint gives \(10\), so the range is \([1,10]\); the excluded input \(0\) does not remove output \(5\), since \(\mathrm f(4)=5\).

For a monotonic branch, track the endpoint outputs in order and transfer open/closed endpoints to the corresponding range boundaries.

Check: on \(3<x\le5\), the same function has range \((2,10]\), not \([1,10]\). The vertex is no longer available.

Ranges on restricted domains

Ranges on restricted domains