Let $$E1=X$$∈ℝ:x≠$$1$$ and xx−$$1$$$$0$$ and $$E2=X$$∈ℝ$$1$$:$$sin$$−$$1logexx$$−$$1is$$ a real number (Here$$, the inverse trigonometric function $$sin$$−$$1x$$ assumes values in−π$$2$$,π$$2$$.Let f:$$E1$$→ℝ be function defined by $$fx=logexx$$−$$1$$ and g:$$E2$$→ℝ be the function defined by $$g(x)=$$$$sin$$−$$1(logxx$$−$$1$$.$$) $$LIST-I$$ $$LIST-II$$ P. The range of f is $$1$$. −∞,$$11$$−e∪ee−$$1$$,∞ Q. The range of g contains $$2$$. (0$$, $$1) R. The domain of f contains $$3$$.. −$$12$$,$$12$$ S. The domain of g is $$4$$.−∞,$$0$$∪$$0$$,∞ $$5$$. −∞,ee−$$1$$ $$6$$. −∞,$$0$$∪$$12$$,ee−$$1$$

Question

Let $$E1=X$$∈ℝ:x≠$$1$$ and xx−$$1$$>$$0$$ and $$E2=X$$∈ℝ$$1$$:$$sin$$−$$1logexx$$−$$1is$$ a real number  (Here$$, the inverse trigonometric function $$sin$$−$$1x$$ assumes values in−π$$2$$,π$$2$$.Let f:$$E1$$→ℝ be function defined by $$fx=logexx$$−$$1$$  and  g:$$E2$$→ℝ be the function defined by $$g(x)=$$$$sin$$−$$1(logxx$$−$$1$$.$$)            $$LIST-I$$                                                             $$LIST-II$$            P. The range of f is                                     $$1$$. −∞,$$11$$−e∪ee−$$1$$,∞           Q. The range of g contains                             $$2$$. (0$$, $$1)            R. The domain of f contains                           $$3$$.. −$$12$$,$$12$$             S. The domain of g is                                   $$4$$.−∞,$$0$$∪$$0$$,∞                                                                                     $$5$$. −∞,ee−$$1$$                                                                                     $$6$$. −∞,$$0$$∪$$12$$,ee−$$1$$

Infinity Learn · Accepted Answer

$$E1$$:$$xx-1$$>$$0$$⇒$$xx-1$$>$$0$$⇒$$E1$$: :x∈$$(-$$∞,$$0)$$∪(1$$,∞$$)E2$$:$$-1$$≤$$lnxx+1$$≤$$11e$$≤$$xx-1$$≤e Now $$xx-1-1e$$≥$$0$$⇒$$(e-1)x+1e(x-1)≥$$0e-1x+1$$≥$$0e-1x$$≥$$-1x$$≤$$11-eand$$ $$xx-1-e$$≤$$0$$ $$x-ex+ex-1$$≤$$0$$ $$x1-e+ex-1$$≤$$0$$ $$x1-e$$≤$$-e$$ x≥$$ee-1Therefore$$, $$E2$$:$$-$$∞,$$11-e$$∪$$ee-1$$,∞

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