Range of a trigonometric function

[terabite22]

Homework Statement

Find the range of the function: y=(2cosx+1)/(2cosx-1) algebraically

Homework Equations

Reducing it, I obtained: y= tan(3x/2)/tan(x/2), but the discontinuity confuses me

The Attempt at a Solution

I did it with my calculator and this is the result:

Ran = (-oo,1/3] U [3,+oo) but I hope I can get help with the algebraic solution.

Thanks in advance.

 

[Office_Shredder]

The discontinuity is quite useful, as that informs you that the range of the function reaches infinity and minus infinity. If cosx=1/2, then x=pi/6(along with a variety of other numbers). 2cos(pi/6)+1=2. So we have something like 2/e, where e is a small number, when x is near pi/6. If x is less than pi/6, e is negative, and can be arbitrarily small. If x is greater than pi/6, e is positive and arbitrarily small. So if e is negative, it goes to negative infinity, if e is positive it goes to positive infinity.

It's similar to how the graph of 1/x works