Question

Given f(x)= 1/1-x and g(x)= x-1/x
find
f of g
g of f^-1
g^-1​

Answer 1

`f(g(x))=x`

`g(f^(-1)(x))=(-1)/(x-1)`

`g^(-1)(x)=(1)/(1-x)`

Explanation:

`f(x)=(1)/(1-x) \ \ \textsf{and} \ \ g(x)=(x-1)/(x)`

`f(g(x))=(1)/(1-((x-1)/(x)))`

`\implies f(g(x))=(1)/((x)/(x)-((x-1))/(x))`

`\implies f(g(x))=(1)/((x-x+1)/(x))`

`\implies f(g(x))=(1)/((1)/(x))`

`\implies f(g(x))=x`

`x=(1)/(1-y)`

`\implies 1-y=(1)/(x)`

`\implies y=1-(1)/(x)`

`\implies f^(-1)(x)=1-(1)/(x)`

`g(f^(-1)(x))=1-(1)/(1-(1)/(x))`

`\implies g(f^(-1)(x))=1-(1)/((x-1)/(x))`

`\implies g(f^(-1)(x))=1-(x)/(x-1)`

`\implies g(f^(-1)(x))=(x-1-x)/(x-1)`

`\implies g(f^(-1)(x))=(-1)/(x-1)`

`x=(y-1)/(y)`

`\implies xy=y-1`

`\implies 1=y-xy`

`\implies 1=y(1-x)`

`\implies y=(1)/(1-x)`

`\implies g^(-1)(x)=(1)/(1-x)`