Question

Domain of function \(f(x) = cos^{-1}\sqrt {2x-1}\) is

• A. [-1,1]
• B. \((\frac{1}{2},1]\)
• C. \([\frac{1}{2},1]\)
• D. \([\frac{1}{2},1)\)

Answer 1

The Correct Option is C

The function to be analyzed is \(f(x) = \cos^{-1}\sqrt{2x-1}\). To determine the domain of \(f(x)\), we start by examining the expression inside the inverse cosine function: \(\sqrt{2x-1}\). The range of the cosine inverse function, \(\cos^{-1}(y)\), is defined for values of \(y\) within \([0, 1]\). This means that for \(\cos^{-1}\sqrt{2x-1}\) to be defined, \(\sqrt{2x-1}\) must satisfy the condition \(0 \leq \sqrt{2x-1} \leq 1\).

To solve this inequality, we square both sides: \(0^2 \leq (2x-1) \leq 1^2\), resulting in:

\[0 \leq 2x-1 \leq 1\]

We solve this compound inequality in two steps:

1. \(2x-1 \geq 0\): Solving gives \(2x \geq 1\), leading to \(x \geq \frac{1}{2}\).
2. \(2x-1 \leq 1\): Solving gives \(2x \leq 2\), leading to \(x \leq 1\).

Combining these results, the solution to the inequality is:

\[\frac{1}{2} \leq x \leq 1\]

Therefore, the domain of the function is the interval \([\frac{1}{2}, 1]\).