Let us begin with the domain.

The domain of y=asin(x) is [-1,1].

Thus, for the function,

f(x)=2asin(3x+4)-1

We require that,

-1<=3x+4<=1

-5<=3x<=-3

-5/3<=x<=-1

Now the range.

To find the range we use my little trick.

"The range of y=f(x) are all y such that the equation y=f(x) has at least one solution for x in the domain".

We begin by writing,

y=2*asin(3x+4)-1

Thus,

y+1=2*asin(3x+4)

Thus,

(y+1)/2=asin(3x+4)

In order to have a solution i.e.

(1/3)sin((y+1)/2)-4=x

We require that,

-pi/2<=(y+1)/2<=pi/2

-pi-1<=y<=pi-1