F(x) = 1/x+5, g(x) = 5x+1/x use composition to prove whether or not the functions are inverses of each other and express the domain of the compositions using interval notation.

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,∞)