Much like a rational function in xx, we must ensure that the denominator is not 00.
3 - cos2x ne 03−cos2x≠0
3 ne cos2x3≠cos2x
Note that the range of the cosine function is [-1, 1][−1,1] so this inequality always holds true. In other words, the denominator is never equal to 00 so the domain is (-oo, oo)(−∞,∞).
FF is a continuous function so it suffices to find the maximum and minimum value for the range. Notice that only the denominator is affected by xx. In this case, we wish to have the greatest and lowest denominator possible to minimize and maximize FF.
cos2x in [color(red)(-1), color(blue)(1)]cos2x∈[−1,1]
The greatest denominator is 3 - (color(red)(-1))3−(−1) which gives a minimum value of:
5/(3 - (color(red)(-1))) = 5/453−(−1)=54
The lowest denominator is 3 - (color(blue)1)3−(1) which gives a maximum value of:
5/(3 - (color(blue)1)) = 5/253−(1)=52
The range of FF must therefore be [5/4, 5/2][54,52].
