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JEE MATHEMATICS MCQ: FUNCTIONS

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Let F:R→RF:R \to R and g:R→Rg:R \to R be two functions defined by f(x)=log⁡e(x2+1)−e−x+1f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1 and g(x)=1−2e2xex.g(x) = \frac{{1 - 2{e^{2x}}}}{{{e^x}}}.
Then for which of the following range of α,\alpha , the inequality f(g((α−1)23))>f(g(α−53))f\left( {g\left( {\frac{{{{(\alpha - 1)}^2}}}{3}} \right)} \right) > f\left( {g\left( {\alpha - \frac{5}{3}} \right)} \right) holds?
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Correct option: A
(2, 3)
Option A ((2, 3)) is correct because by determining that f(x) is increasing and g(x) is decreasing, the inequality can be simplified to a quadratic inequality, which gives the solution (2, 3).

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