The given functions are
[tex] f(x) = \frac{3}{x-5} , g(x) = \frac{3+5x}{x} [/tex]
First we find fog(x)= f(g(x))
So we need to substitute the value of g(x) for x in f(x), that is
[tex] f(g(x)) = \frac{3}{\frac{3+5x}{x}-5} = \frac{3x}{3+5x-5x} [/tex]
[tex] = \frac{3x}{3} = x [/tex]
Now we need to check the value of gof(x)
gof(x) = g(f(x))
So we need to substitute the value of f(x) in g(x), that is
[tex] g(f(x))=\frac{3+5*\frac{3}{x-5}}{\frac{3}{x-5}}= \frac{3x-15+15}{3} [/tex]
[tex] =\frac{3x}{3} = x [/tex]
And since
[tex] fog(x) =gof(x) =x [/tex]
So the functions are inverse of each other .
Correct option is C .
