Question 62544
Find the inverse function for f(x)=4x-2/3x.  Check your answer.
Let f(x)=y
{{{y=(4x-2)/3x}}}
Change the y's to x's and  the x's to y's.
{{{x=(4y-2)/3y}}}  Solve for y.
{{{3y(x)=(3y)(4y-2)/3y}}}
{{{3xy=4y-2}}}
{{{3xy-4y=-4y+4y-2}}}
{{{3xy-4y=-2}}}
{{{y(3x-4)=-2}}}
{{{y(3x-4)/(3x-4)=-2/(3x-4)}}}
{{{y=-2/(3x-4)}}}
let y=f^-1(x)
f^-1(x)={{{-2/(3x-4)}}}
To check take the f(f^-1(x)) and f^-1(f(x)) if they equal x, then they are inverses. [This is taking the composite functions.  You substitute the functions into each other and simplify.  If this is completely foreign to you let me know and I'll explain this more thoroughly.]
f(f^-1(x))={{{(4(-2/(3x-4))-2)/(3(-2/(3x-4)))}}}
={{{(-8/(3x-4)-2)/(-6/(3x-4))}}}
={{{(3x-4)(-8/(3x-4)-2)/((3x-4)(-6/(3x-4)))}}}
{{{(-8-2(3x-4))/-6}}}
{{{(-8-6x+8)/-6}}}
{{{-6x/-6}}}
{{{x}}}  Yes!!!! This means that they are very likely inverses, but to be thorough, check f^-1(f(x))
f^-1(f(x))={{{-2/(3((4x-2)/3x)-4)}}}
={{{-2/((4x-2)/x-4)}}}
={{{-2x/(x(4x-2)/x-4))}}}
={{{-2x/(4x-2-4x)}}}
={{{-2x/-2}}}
={{{x}}} Yes!!! This means that we're right!!!!
Happy Calculating!!!