The sine function for all values gives a result that lies in [-1, +1]

For f(x) = arc sin (2x - 3), the domain is the values of x for which the value of f(x) is defined.

So we get -1 =< 2x - 3 =< 1

=> 2 =< 2x

=> 1=< x

and 2x - 3 =< 1

=> 2x =< 4

=> x =< 2

**The domain of f(x) is [1 , 2]**

To determine the maximum domain of definition of the function, w'ell impose constraints to the argument 2x - 3.

We know that the domain of definition of the arcsine function is [-1 ; 1], so the argument of the function must belong to this range of values.

-1 =< 2x - 3 =< 1

We'll solve the left inequality:

2x - 3 >= -1

2x >= 3 - 1

2x >= 2

x >= 1

We'll solve the right inequality:

2x - 3 =< 1

2x =< 3+1

2x =< 4

x =< 2

**The maximum domain of the given function f(x)=arcsin(2x - 3), is: [1 ; 2].**