JEE Main 2021 (Online) 24th February Morning ShiftQuestion 1
Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=2x-1$ and $g:\mathbf{R}-\{1\}\to\mathbf{R}$ be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$. Then the composition function $f(g(x))$ is :
Functions
Solution
$f(g(x))=\frac{2x-1}{x-1}-1=\frac{x}{x-1}=1+\frac1{x-1}$ on $\mathbb R-\{1\}$: strictly monotone on each branch and injective overall, but never equals $1$, so not onto $\mathbb R$.