In calculus, you can find the domain by analyzing the derivatives of your function.
First, find and set it equal to zero to find all local maxima and minima. A first derivative test will tell you the function is decreasing from x=negative infinity to x=-3 (where it hits a minimum at (-3,-0.2)), then increasing untiil x=1 (there is a discontinuity at x=1), then increases until x=1.8 (where there is a maximum at (1.8,-5)), then decreases until x=3 (there is a disconuity at x=3), then decreases over the remainder of its domain.
Can you picture this in your mind? Or even draw a quick sketch of this behavior?
The only real question is whether the functional values pass between -0.2 and -5 when it is decreasing as x gets larger. Looking at the original function there is a horizontal asymptote at y=0, so it can't pass through these values.
You can go through the same analysis of the second derivative to determine the intervals where the function is concave up and down. But the first derivative test will really tell you what you need to know for this function.
The range is -0.2<=y<=-5.