Let f, g:(1, ) R be defined as f(x)=(2 x+3)/(5 x+2) and g(x)=(2-3 x)/(1-x).

Question

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

Accepted Answer

$$\begin{aligned}
& g(2)=4, g(4)=\frac{10}{3} \
& f \text { is decreasing in }\left(\frac{10}{3}, 4\right) \
& \therefore \quad \alpha=f(4)=\frac{1}{2} \
& \beta=f\left(\frac{10}{3}\right)=\frac{29}{56} \
& \frac{1}{\beta-\alpha}=\frac{1}{\frac{29}{56}-\frac{1}{2}}=56
\end{aligned}$$

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