COMEDK 2023 Shift 1 Paper

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≤1
To solve the inequality:
Add 1 to each part of the inequality:
‌−1+1≤3x−1+1≤1+1
‌0≤3x≤2
Now divide the entire inequality by 3 :
‌‌

0

3

≤‌

3x

3

≤‌

2

3

‌0≤x≤‌

2

3

Thus, the values of x satisfying 0≤x≤‌

2

3

are the domain of f(x).
This can be written in interval notation as:
[0,‌

2

3

]
Therefore, the correct answer is Option A: [0,‌

2

3

].