The inverse cosine function cos−1 only accepts inputs between −1 and 1 (inclusive).
For cos−1(2x−3) to be defined, the input must satisfy:
−1≤2x−3≤1
From the left inequality:
−1≤2x−3
−1+3≤2x
2≤2x
1≤x
From the right inequality:
2x−3≤1
2x≤1+3
2x≤4
x≤2
Combining both results:
1≤x≤2
Therefore, the domain is [1,2].