The given function is:
[tex]f(x)=\frac{2x}{2+3x}[/tex]Graph the function as shown below:
As seen from the horizontal line test, the lines cut the function at only 1 point.
Hence the function is 1-1 and therefore invertible.
[tex]f(x)=y=\frac{2x}{2+3x}[/tex]Solve for x to get:
[tex]\begin{gathered} 2y+3xy=2x \\ 2y=2x-3xy \\ 2y=x(2-3y) \\ x=\frac{2y}{2-3y} \end{gathered}[/tex][tex]\text{ Since y=f(x),x=f}^{-1}(y)[/tex]Therefore replace y by x to get the required inverse function shown below:
[tex]\begin{gathered} x=f^{-1}(y)=\frac{2y}{2-3y} \\ f^{-1}(x)=\frac{2x}{2-3x} \end{gathered}[/tex]The inverse function is shown above.
