[tex]\text{We have been given that}\\ f(x)=\frac{1}{x+5}\\ \\ g(x)=\frac{2}{x-5}\\ \\ \text{We need to find the composition f(g(x)) and g(f(x))}\\ \text{Let us find these one by one}\\[/tex]
[tex]f(g(x))= \frac{1}{\frac{2}{x-5}+5}[/tex]
[tex]f(g(x))=\frac{1}{\frac{2+5x-25}{x-5}} \\ \\ f(g(x))=\frac{x-5}{5x-23}\\[/tex]
[tex]\text{Now, we find second composition function}\\ \\ g(f(x))= \frac{2}{\frac{1}{x+5}-5}\\ \\ \\ g(f(x))=\frac{2}{\frac{1-5x-25}{x+5}}\\ \\ \\ g(f(x))=\frac{2(x+5)}{-5x-24}\\ \\ g(f(x))=-\frac{2(x+5)}{5x+24}[/tex]
