# What is the domain and range of F(x) = 5/(3-cos2x)

What is the domain and range of $F\left(x\right)=\frac{5}{3-\mathrm{cos}2x}$?
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Rayna Aguilar
Much like a rational function in x, we must ensure that the denominator is not 0.
$3-\mathrm{cos}2x\ne 0$
$3\ne \mathrm{cos}2x$
Note that the range of the cosine function is [-1,1] so this inequality always holds true. In other words, the denominator is never equal to 0 so the domain is $\left(-\mathrm{\infty },\mathrm{\infty }\right)$
F 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 x. In this case, we wish to have the greatest and lowest denominator possible to minimize and maximize F.
$\mathrm{cos}2x\in \left[-1,1\right]$
The greatest denominator is 3-(-1) which gives a minimum value of:
$\frac{5}{3-\left(-1\right)}=\frac{5}{4}$
The lowest denominator is 3-(1) which gives a maximum value of:
$\frac{5}{3-\left(1\right)}=\frac{5}{2}$
The range of F must therefore be $\left[\frac{5}{4},\frac{5}{2}\right]$