JEE Main 2025 (Online) 4th April Morning ShiftQuestion 1
Let $f,g:(1,\infty)\to\mathbb{R}$ be defined as $f(x)=\frac{2x+3}{5x+2}$ and $g(x)=\frac{2-3x}{1-x}$. If the range of the function $f\circ g:[2,4]\to\mathbb{R}$ is $[\alpha,\beta]$, then $\frac{1}{\beta-\alpha}$ is equal to
Functions
Solution
$g(x)=3+\frac1{x-1}$ maps $[2,4]$ onto $\left[\frac{10}3,4\right]$; $f(t)=\frac{2t+3}{5t+2}$ is decreasing, so $f\circ g$ has range $\left[f(4),f\left(\frac{10}3\right)\right]=\left[\frac12,\frac{29}{56}\right]$. Then $\frac1{\beta-\alpha}=56$.