JEE Main 2022 (Online) 28th July Morning ShiftQuestion 1
Let $\alpha,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x$, $x\in\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be such that $g(f(x))=x$ for all $x\in\mathbf{R}$. If $a_1,a_2,a_3,\ldots,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac1n\sum_{i=1}^{n}f(a_i)\right)\right)$ is equal to :
Sets, Relations and Function
Solution
$f$ is odd and strictly increasing (all coefficients positive), so $g=f^{-1}$ is well defined and odd. An AP with mean zero is symmetric: $a_i=-a_{n+1-i}$, so $\sum f(a_i)=0$ by oddness. Hence $f\left(g(0)\right)=0$. Option (A).