Reversed Composite Function

Featured problem for today was taken from EMC blitz Quiz organized by one of the members of the active group in facebook called Elite Math Circle. It is about composite function but the problem is reversed. It is very easy at first glance but a bit hard and confusing to solve.

 

 

Problem:

If (f\circ g)(x)=\displaystyle\frac{1}{x-25} and f(x)=x^2+5, find g(x).

Solution:

(f\circ g)(x)=f(g(x)) \displaystyle\frac{1}{x-25}=(g(x))^2+5 (g(x))^2=\displaystyle\frac{1}{x-25}-5 (g(x))^2=\displaystyle\frac{1-5(x-25)}{x-25} (g(x))^2=\displaystyle\frac{-5x+126}{x-25}

Extracting the squares of both sides,

g(x)=\sqrt{\displaystyle\frac{-5x+126}{x-25}}

 

 

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