Question 838748
To find the inverse we can solve for x to convert it to the dependent variable:

{{{y=( 2x )/(3x-5)}}}

{{{(3x-5)y= 2x }}}


{{{3xy-5y= 2x }}}


{{{3xy-2x=  5y}}}


{{{x(3y-2)=  5y}}}


{{{x=  (5y)/(3y-2)}}}

Once their roles have reversed, the resulting expression is the inverse:

{{{g(x)=(5x)/(3x-2)}}}

To check we take the two appropriate inverses to see if they each recover the input (i.e. x).


Checking...

{{{f(g(x))=(2((5x)/(3x-2)))/(3((5x)/(3x-2))-5)}}}


{{{f(g(x))=((10x)/(3x-2))/((15x)/(3x-2)-5*(3x-2)/(3x-2))}}}



{{{f(g(x))=((10x)/(3x-2))*((3x-2)/(15x-15x+10))=x}}}


{{{g(f(x))=(5((2x)/(3x-5)))/(3((2x)/(3x-5))-2)=((10x)/(3x-5))((3x-5)/(6x-6x+10))=x}}}