Note that the denominator is undefined whenever
4 sin(x) + 2 = 0,
that is, whenever
x = x_(1,n) = pi/6 + n 2pi
or
x = x_(2,n) = (5 pi)/6 + n 2pi,
where n in ZZ (n is an integer).
As x approaches x_(1,n) from below, f(x) approaches - infty, while if x approaches x_(1,n) from above then f(x) 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(x) approaches +infty, while if x approaches x_(2,n) from above then f(x) approaches -infty.
We get a sequence of intervals in which f(x) 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(x) 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(x) is as large as possible, that is when sin(x) = 1. So we conclude that the smallest positive value of f(x) is 1/(4*1 + 2) = 1/6.
The largest negative value is similarly found to be 1/(4*(-1) + 2) = -1/2.
Due to the continuity of f(x) in the intervals between discontinuities, and the Intermediate value theorem , we can conclude that the range of f(x) is
R = (-infty, -1/2] uu [1/6, +infty)
The hard brackets mean that the number is included in the interval (e.g. -1/2), while soft brackets means that the number is not included.
graph{1/(4sin(x) + 2) [-10, 10, -5, 5]}