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Mathematics Question on Sets

The domain of sin⁡−1[log⁡3(x3)]\sin^{-1}\left[\log_3\left(\frac{x}{3}\right)\right] is

A

[1, 9]

B

[- 1, 9]

C

[- 9, 1]

D

[-9, -1]

Answer

[1, 9]

Explanation

Solution

f(x)=sin⁡−1(log3(x3))f(x)= \sin^{-1}\left(log_3\left(\frac{x}{3}\right)\right) is defined if −1≤ log⁡3(x3)≤ 1⇒3−1≤x3≤31-1 \leq \,\log_3 \left(\frac{x}{3}\right)\leq \,1 \Rightarrow 3^{-1} \leq \frac{x}{3}\leq 3^1 ⇒13≤x3≤3⇒1≤x ≤9,∴Df=[1,9]\Rightarrow \frac{1}{3} \leq \frac{x}{3} \leq 3 \Rightarrow 1 \leq x \,\leq 9 , \therefore D_f=[1,9]