[tex]\begin{gathered} \text{Given} \\ k(x)=\frac{2x}{3-4x} \end{gathered}[/tex][tex]\begin{gathered} k(x)=\frac{2x}{3-4x} \\ y=\frac{2x}{3-4x} \end{gathered}[/tex]
Convert y to x, and x to y
[tex]\begin{gathered} y=\frac{2x}{3-4x} \\ x=\frac{2y}{3-4y} \end{gathered}[/tex]
Solve for y.
[tex]\begin{gathered} x=\frac{2y}{3-4y} \\ 3-4y=\frac{2y}{x} \\ \frac{3-4y}{2y}=\frac{\frac{2y}{x}}{2y} \\ \frac{3}{2y}-\frac{4y}{2y}=\frac{1}{x} \\ \frac{3}{2y}-2=\frac{1}{x} \\ \frac{3}{2y}=\frac{1}{x}+2 \end{gathered}[/tex]
Get the reciprocal of both sides.
[tex]\begin{gathered} \mleft(\frac{3}{2y}=\frac{1}{x}+2\mright)^{-1} \\ (\frac{3}{2y}=\frac{1+2x}{x})^{-1} \\ \frac{2y}{3}=\frac{x}{1+2x} \\ \text{Multiply both sides by }\frac{3}{2}\text{ to cancel out }\frac{2}{3}\text{ on the left side} \\ \frac{3}{2}\mleft(\frac{2y}{3}=\frac{x}{1+2x}\mright)\frac{3}{2} \\ y=\frac{3x}{2(1+2x)} \\ \\ \text{Simplify the denominator} \\ y=\frac{3x}{2+4x} \\ \text{rearrange the terms on the denominator} \\ y=\frac{3x}{4x+2} \end{gathered}[/tex][tex]\begin{gathered} \text{Therefore,} \\ k^{-1}(x)=\frac{3x}{4x+2} \end{gathered}[/tex]
