If f(x) = 2x/x+1, what is f^-1(x)?

Write $y=\frac{2x}{x+1}$ and swap the variables: $x=\frac{2y}{y+1}$. Multiply both sides by $y+1$: $xy+x=2y$. Gather the $y$ terms: $x=2y-xy=y(2-x)$. Divide: $y=\frac{x}{2-x}$, choice F. Check with a value: $f(1)=\frac{2}{2}=1$, and $f^{-1}(1)=\frac{1}{2-1}=1$. Choice J ($\frac{x+1}{2x}$) is the reciprocal of the original function, which is not an inverse. Choice G ($\frac{x}{x+2}$) has a sign error in the denominator. Choice H ($\frac{2-x}{x}$) is the reciprocal of the correct answer. Verifying an inverse by testing a value is a reliable shortcut.