Finding the range of f(x)=sin2x+sin⁡x−1sin2x−sin⁡x+2 I am attempting to find the range of the function f→R→R which is given by the expression as follows. f(x)=sin2x+sin⁡x−1sin2x−sin⁡x+2 This is rational function in sin⁡x, so it will be periodic. Usually when there is a rational function with quadratic expressions, differentiation can be used or, x is rewritten in terms of y, so that gives. sin⁡x=(y+1)±10−2y−3y22(y−1)∈[−1,1] This is a tedious task and so is using differentiation to find minima and maxima. Any hints to go about this problem using a different technique are appreciated.

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Finding the range of f(x)=sin2x+sin⁡x−1sin2x−sin⁡x+2  I am attempting to find the range of the function f→R→R which is given by the expression as follows.  f(x)=sin2x+sin⁡x−1sin2x−sin⁡x+2  This is rational function in sin⁡x, so it will be periodic. Usually when there is a rational function with quadratic expressions, differentiation can be used or, x is rewritten in terms of y, so that gives.  sin⁡x=(y+1)±10−2y−3y22(y−1)∈[−1,1]  This is a tedious task and so is using differentiation to find minima and maxima. Any hints to go about this problem using a different technique are appreciated.

RizerMix · Accepted Answer

Let φ(x)=x2+x−1x2−x+2 then f=φ⋅sin. Since the range of sin⁡ is [−1,1], the range of f is φ([−1,1]). But φ is stricly decreasing on [−1,3112] and strictly increasing on [3−112,1]. Since, f(−1)=−14 f(3112)=3−2117 f(1)=12 the range is [3−2117,12]

Finding the range of f(x)=\frac{\sin^2 x+\sin x-1}{\sin^2 x-\sin x+2} I am attempting to f

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