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

Question

Let $f,g:(1,\infty)	o R$ be defined as $f(x)=\frac{2x+3}{5x+2}$ and $g(x)=\frac{2-3x}{1-x}$. If the range of the function fog: $[2,4]	o R$ is $[\alpha ,\beta ]$, then $\frac{1}{\beta -\alpha }$ is equal to

Accepted Answer

$\begin{gathered} g(2)=4,g(4)=\frac{10}{3} \ f(\frac{10}{3},4) \ 	herefore \, \alpha =f(4)=\frac{1}{2} \ \beta =f(\frac{10}{3})=\frac{29}{56} \ \frac{1}{\beta -\alpha }=\frac{1}{\frac{29}{56}-\frac{1}{2}}=56 \end{gathered}$

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