Question

Solve both f(g(x)) and g(f(x))

if f(x)= 1/x+5 and g(x)=2/x-5

Please help me I am struggling with this and it would be greatly appreciated.

Answer 1

`\text{We have been given that}\\ f(x)=(1)/(x+5)\\ \\ g(x)=(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}\\`

`f(g(x))= (1)/((2)/(x-5)+5)`

`f(g(x))=(1)/((2+5x-25)/(x-5)) \\ \\ f(g(x))=(x-5)/(5x-23)\\`

`\text{Now, we find second composition function}\\ \\ g(f(x))= (2)/((1)/(x+5)-5)\\ \\ \\ g(f(x))=(2)/((1-5x-25)/(x+5))\\ \\ \\ g(f(x))=(2(x+5))/(-5x-24)\\ \\ g(f(x))=-(2(x+5))/(5x+24)`