Embibe Experts Solutions for Chapter: Functions, Exercise 1: JEE Advanced Paper 2 - 2020

Let the function $f:\left[0,1\right]\to \mathrm{ℝ}$ be defined by $f\left(x\right)=\frac{4^x}{4^x+2}$. Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\cdots +f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$  is_________

If the function $f:R\to R$ is defined by $f\left(x\right)=\left|x\right|\left(x-\sin x\right)$, then which of the following statements is TRUE?

Let $f\left(x\right)=\sin \left(\frac{\pi }{6}\sin \left(\frac{\pi }{2}\sin x\right)\right)$, for all $x\in R$ and $g\left(x\right)=\frac{\pi }{2}\sin x$, for all $x\in R$. Let $\left(fog\right)\left(x\right)$ denote $f\left(g\left(x\right)\right)$ and $\left(gof\right)\left(x\right)$ denote $g\left(f\left(x\right)\right)$. Then which of the following is (are) true?

Let $f :\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to R$ be given by $f\left(x\right)={\left(\log \left(\sec x+\tan x\right)\right)}^3.$ Then