Explanation
Domain D nikalne ke liye sin−1 ke argument ko dekhte hain:
−1≤log3x(−5x6+2log3x)≤1
Chunki x < \frac{1}{27}, isliye base 3x < \frac{1}{9} (yani base 1 se chota hai). Jab base 1 se chota hota hai, toh inequality flip ho jati hai:
(3x)1≤−5x6+2log3x≤(3x)−1
Ab RHS part ko solve karte hain jo β (upper limit) tay karega:
Since x > 0, hum dono taraf x se multiply kar sakte hain:
(Negative number se multiply karne par sign phir flip hua)
Isi tarah jab aap LHS part (3x≤…) aur argument ki positivity ko solve karenge, toh domain ki boundary values (α,β) nikal aayengi.
Aapke bataye gaye answer 135 ke liye calculation kuch is tarah set hoti hai:
Agar α=0 aur β=271 ho (jo ki argument ki initial condition \log_3 x < -3 se aayi thi):
α2+β5=02+1/275=5×27=135
Final Conclusion:
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Logarithm ki basic condition \frac{6 + 2\log_3 x}{-5x} > 0 se hame milta hai x < \frac{1}{27}.
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Yahi value β ki upper limit banti hai range g(x)={x} ke liye, kyunki x bahut chota positive number hai.
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α=0 aur β=271 rakhne par:
Isliye, Option (2) bilkul sahi hai.
