Question

Find the domain of the function \( f(x)=\sin^{-1(3x-1) \).}

• A. \( [-1,1] \)
• B. \( [0,\frac{2}{3}] \)
• C. \( [-\frac13,\frac13] \)
• D. \( [\frac13,1] \)

Hint:

For inverse trigonometric functions, always restrict the inner expression according to the valid range of the original trigonometric function.

Answer 1

The Correct Option is B

Concept: The inverse sine function is defined only when its argument lies in the interval: \[ -1 \le u \le 1 \]

Step 1: Apply the inverse sine condition:
For \[ f(x)=\sin^{-1}(3x-1) \] we require: \[ -1 \le 3x-1 \le 1 \]

Step 2: Solve the inequality:
Add \(1\) throughout: \[ 0 \le 3x \le 2 \] Divide by \(3\): \[ 0 \le x \le \frac23 \]

Step 3: Write the domain:
Hence the domain is: \[ \boxed{\left[0,\frac23\right]} \]