Amplitude and period describe the whole periodic function. Its range on a restricted domain is the set of heights actually reached by allowed inputs. A continuous sine or cosine segment reaches its extreme height either at an included turning point or at a domain endpoint. An excluded endpoint contributes a limiting height, but that height might still occur at another included input. Map the input interval through bx; locate the included maxima/minima; calculate endpoint heights; choose smallest/largest heights and decide whether each is attained. For tangent split at every asymptote and combine the separate branch ranges. For f(x)=2sin x+1, 0 <= x<pi/2, the curve rises from 1 towards 3, so the range is [1,3). For -pi/2<x<pi/2, f(0)=1 is included even though the endpoints are not. For g(x)=tan x, 0 <= x 3pi/4 with x!= pi/2, the first branch gives [0,infinity) and the second gives (-infinity,-1]. Do not join across the asymptote or claim every real height is reached.

Amplitude and period describe the whole periodic function. Its range on a restricted domain is the set of heights actually reached by allowed inputs.

A continuous sine or cosine segment reaches its extreme height either at an included turning point or at a domain endpoint. An excluded endpoint contributes a limiting height, but that height might still occur at another included input.

Map the input interval through bx; locate the included maxima/minima; calculate endpoint heights; choose smallest/largest heights and decide whether each is attained. For tangent split at every asymptote and combine the separate branch ranges.

For \(f(x)=2\sin x+1\), \(0\le x<\frac{\pi}{2}\), the curve rises from 1 towards 3, so the range is \([1,3)\). For \(-\frac{\pi}{2}<x<\frac{\pi}{2}\), \(f(0)=1\) is included even though the endpoints are not.

For \(g(x)=\tan x\), \(0\le x\le\frac{3\pi}{4}\) with \(x\ne\frac{\pi}{2}\), the first branch gives \([0,\infty)\) and the second gives \((-\infty,-1]\). Do not join across the asymptote or claim every real height is reached.

Ranges on restricted trigonometric domains

Ranges on restricted trigonometric domains