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

Aug 3, 2018

The domain is $\left(- \infty , \infty\right)$. The range is $\left[\frac{5}{4} , \frac{5}{2}\right]$.

#### Explanation:

Much like a rational function in $x$, we must ensure that the denominator is not $0$.

$3 - \cos 2 x \ne 0$

$3 \ne \cos 2 x$

Note that the range of the cosine function is $\left[- 1 , 1\right]$ so this inequality always holds true. In other words, the denominator is never equal to $0$ so the domain is $\left(- \infty , \infty\right)$.

$F$ 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 $F$.

$\cos 2 x \in \left[\textcolor{red}{- 1} , \textcolor{b l u e}{1}\right]$

The greatest denominator is $3 - \left(\textcolor{red}{- 1}\right)$ which gives a minimum value of:

$\frac{5}{3 - \left(\textcolor{red}{- 1}\right)} = \frac{5}{4}$

The lowest denominator is $3 - \left(\textcolor{b l u e}{1}\right)$ which gives a maximum value of:

$\frac{5}{3 - \left(\textcolor{b l u e}{1}\right)} = \frac{5}{2}$

The range of $F$ must therefore be $\left[\frac{5}{4} , \frac{5}{2}\right]$.