Question 884712
see the attached worksheet.
step 1 is the original equation.
step 2 interchanges the y and the x.
step 3 solves for y in the interchanged equation.
you get:
f(x) = (4x-2) / (3x + 2) which is the original equation of f(x).
g(x) = (-2x-2) / (3x-4) which is the inverse equation.
you wind up with:
g(x) = f^-1(x) and f(x) = g^-1(x).
this is because:
if a is the inverse of b then b is the inverse of a.


step 4 shows that f(g(x)) is equal to x.
step 5 shows that g(f(x)) is equal to x.


the method of solving f(g(x)) and g(f(x) is to multiply numerator and denominator by the common denominator which removes the fractions and allows you to solve the equations easier.


that's why step 4 equation is multiplied by (3x-4) / (3x-4) and why step 5 equation is multiplied by (3x + 2) / (3x + 2).


see below the pictures for additional comments.

<img src = "http://theo.x10hosting.com/2014/jun2802.jpg" alt="$$$" </>
<img src = "http://theo.x10hosting.com/2014/jun2803.jpg" alt="$$$" </>


you get f(g(x)) = x and g(f(x)) = x which confirms that the equations are inverses of each other.


the general idea to find the inverse equatgion is to do steps 1, 2 and 3.
you replace x with y and y with x and then you solve for  y.
that's what was done in step 3.