What is the maximum domain of definition of function f(x)=arcsin(2x-3)?

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].