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