Question:
The domain of \[ f(x)=\cos^{-1}(2x-5) \] is:
The domain of \[ f(x)=\cos^{-1}(2x-5) \] is:
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Always remember the standard domains of inverse trigonometric functions: \(\sin^{-1}(x)\) and \(\cos^{-1}(x)\) are both defined for \([-1, 1]\).
When solving double inequalities, perform the same operation on the left, middle, and right sides simultaneously.
Updated On: Sep 11, 2026
- \([-1, 1]\)
- \([4, 6]\)
- \([-7, -3]\)
- \([2, 3]\)
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The Correct Option is D
Solution and Explanation
Concept:
• The inverse cosine function, \(y = \cos^{-1}(u)\), has a defined domain of \([-1, 1]\).
• This implies that the argument \(u\) must satisfy the inequality \(-1 \leq u \leq 1\).
• To find the domain of the given function, we solve this inequality for \(x\).
Step 1: Identify the argument and set up the inequality
In the function \(f(x) = \cos^{-1}(2x - 5)\), the argument is \(2x - 5\). For the function to be well-defined, the argument must lie within the interval \([-1, 1]\): \[ -1 \leq 2x - 5 \leq 1 \]
Step 2: Solve the linear inequality for \(x\)
To isolate the term with \(x\), we first add \(5\) to all parts of the inequality: \[ -1 + 5 \leq 2x - 5 + 5 \leq 1 + 5 \] \[ 4 \leq 2x \leq 6 \] Now, divide the entire inequality by \(2\) to solve for \(x\): \[ \frac{4}{2} \leq \frac{2x}{2} \leq \frac{6}{2} \] \[ 2 \leq x \leq 3 \]
Step 3: Express the domain in interval notation
The possible values for \(x\) range from \(2\) to \(3\), including both endpoints. Therefore, the domain is \([2, 3]\).
• The inverse cosine function, \(y = \cos^{-1}(u)\), has a defined domain of \([-1, 1]\).
• This implies that the argument \(u\) must satisfy the inequality \(-1 \leq u \leq 1\).
• To find the domain of the given function, we solve this inequality for \(x\).
Step 1: Identify the argument and set up the inequality
In the function \(f(x) = \cos^{-1}(2x - 5)\), the argument is \(2x - 5\). For the function to be well-defined, the argument must lie within the interval \([-1, 1]\): \[ -1 \leq 2x - 5 \leq 1 \]
Step 2: Solve the linear inequality for \(x\)
To isolate the term with \(x\), we first add \(5\) to all parts of the inequality: \[ -1 + 5 \leq 2x - 5 + 5 \leq 1 + 5 \] \[ 4 \leq 2x \leq 6 \] Now, divide the entire inequality by \(2\) to solve for \(x\): \[ \frac{4}{2} \leq \frac{2x}{2} \leq \frac{6}{2} \] \[ 2 \leq x \leq 3 \]
Step 3: Express the domain in interval notation
The possible values for \(x\) range from \(2\) to \(3\), including both endpoints. Therefore, the domain is \([2, 3]\).
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