JEE Main 2026 (Online) 4th April Evening ShiftQuestion 1
For the function $f:[1,\infty)\to[1,\infty)$ defined by $f(x)=(x-1)^{4}+1$, among the two statements:
(I) The set $S=\{x\in[1,\infty):f(x)=f^{-1}(x)\}$ contains exactly two elements, and
(II) The set $S=\{x\in[1,\infty):f(x)=f^{-1}(x+1)\}$ is an empty set,
Functions
Solution
$f$ is increasing, $f^{-1}(x)=1+(x-1)^{1/4}$. (I) $f(x)=x$: $t=x-1$, $t^4=t$, $t=0,1$, so $x=1,2$: two elements, TRUE. (II) $(x-1)^4=x^{1/4}$ is $0<1$ at $x=1$ and large for big $x$, so a root exists: FALSE.