The function given is
f(x)=cos−1(3x−1).
To determine the domain of
f(x), we need to find the values of
x for which the expression
3x−1 lies within the valid range for the cosine inverse function. The range of
cos−1(y) is
y∈[−1,1], meaning
cos−1(y) is defined when
y is between -1 and 1 (inclusive).
Therefore, to find the domain of
f(x), we need to solve the inequality:
−1≤3x−1≤1To solve the inequality:
Add 1 to each part of the inequality:
‌−1+1≤3x−1+1≤1+1‌0≤3x≤2Now divide the entire inequality by 3 :
‌‌≤‌≤‌‌0≤x≤‌Thus, the values of
x satisfying
0≤x≤‌ are the domain of
f(x).
This can be written in interval notation as:
[0,‌]Therefore, the correct answer is Option A:
[0,‌].