Starting with the expression:
[tex]f(x)=\frac{2}{1+2x}[/tex]Substitute f(x)=y and isolate x:
[tex]\begin{gathered} y=\frac{2}{1+2x} \\ \Rightarrow1+2x=\frac{2}{y} \\ \Rightarrow2x=\frac{2}{y}-1 \\ \Rightarrow x=\frac{1}{y}-\frac{1}{2} \end{gathered}[/tex]Swap x for y and vice versa:
[tex]y=\frac{1}{x}-\frac{1}{2}[/tex]Set y=f^{-1}(x):
[tex]f^{-1}(x)=\frac{1}{x}-\frac{1}{2}[/tex]