005 Sample Final A, Question 4

Question Find the inverse of the following function $f(x)={\frac {3x}{2x-1}}$

Foundations:
1) How would you find the inverse for a simpler function like $f(x)=3x+5$?
Answer:
1) you would replace f(x) by y, switch x and y, and finally solve for y.

Step 1:
Switch f(x) for y, to get $y={\frac {3x}{2x-1}}$, then switch y and x to get $x={\frac {3y}{2y-1}}$

Step 2:
Now we have to solve for y: ${\begin{array}{rcl}x&=&{\frac {3y}{2y-1}}\\x(2y-1)&=&3y\\2xy-x&=&3y\\2xy-3y&=&x\\y(2x-3)&=&x\\y&=&{\frac {x}{2x-3}}\end{array}}$

Final Answer:
$y={\frac {x}{2x-3}}$