The function is one-one can be dtermined by using horizontal line test. If horizontal line on the graph of function intersect the function more than once then such function is not one-one. The graph of function is,
Since horizontal lines intersect the curve of function only once so function is one-one.
Determine the inverse of the function.
[tex]y=\frac{2x}{2+3x}[/tex]
Interchange x with y and y with x and simplify the obtain equation for x.
[tex]\begin{gathered} y=\frac{2x}{2+3x} \\ y\cdot(2+3x)=2x \\ 2y+3xy=2x \\ 3xy-2x=-2y \\ x(3y-2)=-2y \\ x=-\frac{2y}{3y-2} \end{gathered}[/tex]
Substitute y by x for the inverse of function.
[tex]f(x)=\frac{-2x}{3x-2}[/tex]
So inverse of the function is -2x/(3x - 2).
So functions f and g are inverse of each other if,
[tex]f(g(x))=g(f(x))=x[/tex]
Check the obtained inverse function by composition.
[tex]\begin{gathered} f^{-1}(-\frac{2x}{3x-2})=\frac{2\cdot(-\frac{2x}{3x-2})}{2+3\cdot(-\frac{2x}{3x-2})} \\ =\frac{-\frac{4x}{3x-2}}{\frac{6x-4-6x}{3x-2}} \\ =-\frac{4x}{-4} \\ =x \end{gathered}[/tex][tex]undefined[/tex]
