we have
f(x) = 1/x+5,
g(x) = 5x+1/x -------> (5x²+1)/x
1) find the f(g(x))
f(g(x))=(1/(5x²+1)/x) +5---- > (x/(5x²+1)) +5---->(x+5*(5x²+1))/(5x²+1)
(x+25x²+5)/(5x²+1)---------- > (25x²+x+5)/(5x²+1)
f(g(x))= (25x²+x+5)/(5x²+1)--------> is not x
therefore
f(x) and g(x) are not inverses of each other
the domain of the composition f(g(x)) is (-∞,∞)
2) find the g(f(x))
g(f(x))=(5(1/x+5,)²+1)/(1/x+5,)------- > (5/x²)+(10/x)+25+(x/(5x+1))
[5*(5x+1)+10x*(5x+1)+25x²*(5x+1)+x³]/[x²*(5x+1)]
the domain of the composition g(f(x)) is (-∞, -1/5) U (-1/5,0) U (0,∞)
