Let be defined as and . If the range of the function is then is equal to [2025]

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Solution

Q.
Let $f, g : (1, \infty) \to R$ 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 $f o g : [2, 4] \to R$ is $[α, β]$ then $\frac{1}{β - α}$ is equal to [2025]

1 56

2 68

3 29

4 2

Ans.
(1)

Given $f, g : (1, \infty) \to R$ , $f (x) = \frac{2 x + 3}{5 x + 2}$ , $g (x) = \frac{2 - 3 x}{1 - x}$

also, we have $f o g : [2, 4] \to R$

Now, $g (2) = \frac{2 - 6}{1 - 2} = 4, g (4) = \frac{2 - 12}{1 - 4} = \frac{10}{3}$

$∴ f (g (2)) = \frac{8 + 3}{20 + 2} = \frac{1}{2}, f (g (4)) = \frac{20 + 9}{50 + 6} = \frac{29}{56}$

i.e., $α = \frac{1}{2} and β = \frac{29}{56}$

Then, $\frac{1}{β - α} = \frac{1}{\frac{29}{56} - \frac{1}{2}} = \frac{1}{\frac{1}{56}} = 56$ .

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