# What is the range of the function f(x)= 1 / ( 4 sin(x) + 2 ) ?

Apr 4, 2017

The range is $R = \left(- \infty , - \frac{1}{2}\right] \cup \left[\frac{1}{6} , + \infty\right)$

#### Explanation:

Note that the denominator is undefined whenever
$4 \sin \left(x\right) + 2 = 0$,
that is, whenever
$x = {x}_{1 , n} = \frac{\pi}{6} + n 2 \pi$
or
$x = {x}_{2 , n} = \frac{5 \pi}{6} + n 2 \pi$,
where $n \in \mathbb{Z}$ ($n$ is an integer).

As $x$ approaches ${x}_{1 , n}$ from below, $f \left(x\right)$ approaches $- \infty$, while if $x$ approaches ${x}_{1 , n}$ from above then $f \left(x\right)$ approaches $+ \infty$. This is due to division by "almost $- 0$ or $+ 0$".

For ${x}_{2 , n}$ the situation is reversed. As $x$ approaches ${x}_{2 , n}$ from below, $f \left(x\right)$ approaches $+ \infty$, while if $x$ approaches ${x}_{2 , n}$ from above then $f \left(x\right)$ approaches $- \infty$.

We get a sequence of intervals in which $f \left(x\right)$ is continuous, as can be seen in the plot. Consider first the "bowls" (at whose ends the function blows up to $+ \infty$). If we can find the local minima in these intervals, then we know that $f \left(x\right)$ assumes all the values between this value and $+ \infty$. We can do the same for "upside-down bowls", or "caps".

We note that the smallest positive value is obtained whenever the denominator in $f \left(x\right)$ is as large as possible, that is when $\sin \left(x\right) = 1$. So we conclude that the smallest positive value of $f \left(x\right)$ is $\frac{1}{4 \cdot 1 + 2} = \frac{1}{6}$.

The largest negative value is similarly found to be $\frac{1}{4 \cdot \left(- 1\right) + 2} = - \frac{1}{2}$.

Due to the continuity of $f \left(x\right)$ in the intervals between discontinuities, and the Intermediate value theorem , we can conclude that the range of $f \left(x\right)$ is

$R = \left(- \infty , - \frac{1}{2}\right] \cup \left[\frac{1}{6} , + \infty\right)$

The hard brackets mean that the number is included in the interval (e.g. $- \frac{1}{2}$), while soft brackets means that the number is not included.

graph{1/(4sin(x) + 2) [-10, 10, -5, 5]}