Let f, g:(1, ∞) rightarrow mathbb R be defined as f(x)=2 x+3/5 x+2 and g(x)=2-3 x/1-x . If the range of the function f ° g:[2,4] rightarrow mathbb R is [α, β] , then 1/β-α is equal to

Mathematics · Functions

JEE Main 2025 — 4 April, Morning Shift — Question 25

Let $f, g:(1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x)=\frac{2 x+3}{5 x+2}$ and $g(x)=\frac{2-3 x}{1-x}$ . If the range of the function

$f \circ g:[2,4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$ , then $\frac{1}{\beta-\alpha}$ is equal to

$g(2)=4, g(4)=\frac{10}{3}$

$f$ is decreasing in $\left(\frac{10}{3}, 4\right)$

$\therefore \quad \alpha=f(4)=\frac{1}{2}$

$\beta=f\left(\frac{10}{3}\right)=\frac{29}{56}$

$\frac{1}{\beta-\alpha}=\frac{1}{\frac{29}{56}-\frac{1}{2}}=56$

Answer key and solution verified before publishing.

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