#### Please Solve RD Sharma Class 12 Chapter Inverse Trigonometric Function Exercise 3.4 Question 3 Subquestion (i) Maths Textbook Solution.

Answer: $\left (-\infty,0\right ] \cup \left [ \frac{2}{3},\infty \right )$
Hint:  $Domain\left ( y^{-1} \right )= Range\left ( y \right )$
$Domain\, o\! f\sec ^{-1}\left ( x \right )=\left ( -\infty,-1 \right ]or\left [ 1,\infty \right ]$
Given:    $\sec ^{-1}\left ( 3x-1 \right )$
Solution:$W\! e\, know \, that\, the\, range\, o\! f\sec x\, is \left ( -\infty,-1 \right )\cup \left ( 1,\infty \right )$
$3x-1< -1\: and \: 3x-1\geq 1$
$3x< -1+1\: and \: 3x\geq 1+1$
$3x< 0 \: and\: 3x\geq 2$
$x< 0\: and\: x\geq \frac{2}{3}$
$x\: \epsilon \: \left ( -\infty,0 \right ]and\: x\: \epsilon \left [ \frac{2}{3},\infty \right ]$
$x\: \epsilon \: \left ( -\infty,0 \right ]\cup \left [ \frac{2}{3},\infty \right ]$
$Domain\, is \left (-\infty,0\right ] \cup \left [ \frac{2}{3},\infty \right ]$

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Hint:

Domain of  or

Given:

Solution: We know that the range of  is

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Domain is