The inverse of a function.

Question

I am unable to reach the inverse function of $$\frac{x}{2x+1}$$

For $y=\frac{x}{2x+1}$ I tried, but I am still unable to match the answer according to my book, which is $$\frac{x}{1-2x}$$

Answer 1

To find the inverse switch $y$ and $x$ then solve for $y$.

Answer 2

$$f\left( x \right) =\frac { 1 }{ 2x+1 } \\ 2x+1=\frac { 1 }{ f\left( x \right) } \\ x=\frac { \frac { 1 }{ f\left( x \right) } -1 }{ 2 } =\frac { 1-f\left( x \right) }{ 2f\left( x \right) } $$ so

$$f^{ -1 }\left( x \right) =\frac { 1-x }{ 2x } $$