1) In order to find the inverse function of that function, let's do it step by step
Given the function, remember f(x) and y is the same thing.
[tex]f(x)=\frac{x}{2x+3}[/tex]
Swap the x for y:
[tex]\begin{gathered} y=\frac{x}{2x+3} \\ x=\frac{y}{2y+3} \\ \text{Cross multiply:} \\ x(2y+3)\text{ =y} \\ 2xy\text{ +3x = y} \\ \text{Subtract y from both sides} \\ 2xy-y+3x=0 \\ Subtract\text{ 3x from both sides} \\ 2xy-y=-3x \\ \text{Write the left side as a factor} \\ y(2x-1)\text{ =-3x} \\ \text{Divide both sides by (2x-1)} \\ y=\frac{-3x}{2x-1} \\ \text{Write the proper notation:} \\ f^{-1}(x)=\frac{-3x}{2x-1} \end{gathered}[/tex]
2) Then the inverse function for that one is:
[tex]f^{-1}(x)=\frac{-3x}{2x-1}[/tex]
