# 004 Sample Final A, Problem 1

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Find ${\displaystyle f^{-1}(x)}$ for ${\displaystyle f(x)={\frac {3x-1}{4x+2}}}$

Foundations
How would you find the inverse for a simpler function like ${\displaystyle f(x)=2x+4?}$
You would replace ${\displaystyle f(x)}$ with ${\displaystyle y}$. Then, switch ${\displaystyle x}$ and ${\displaystyle y}$. Finally, we would solve for ${\displaystyle y}$.

Solution:

Step 1:
We start by replacing ${\displaystyle f(x)}$ with ${\displaystyle y}$.
This leaves us with ${\displaystyle y={\frac {3x-1}{4x+2}}}$
Step 2:
Now, we swap ${\displaystyle x}$ and ${\displaystyle y}$ to get ${\displaystyle x={\frac {3y-1}{4y+2}}}$.
Step 3:
Starting with ${\displaystyle x={\frac {3y-1}{4y+2}}}$, we multiply both sides by ${\displaystyle 4y+2}$ to get
${\displaystyle x(4y+2)=3y-1}$.
Now, we need to get all the ${\displaystyle y}$ terms on one side. So, adding ${\displaystyle 1}$ and ${\displaystyle -4xy}$ to both sides we get
${\displaystyle 2x+1=3y-4xy}$.
Step 4:
Factoring out ${\displaystyle y}$, we get ${\displaystyle 2x+1=y(3-4x)}$. Now, dividing by ${\displaystyle (3-4x)}$, we get
${\displaystyle {\frac {2x+1}{3-4x}}=y}$. Replacing ${\displaystyle y}$ with ${\displaystyle f^{-1}(x)}$, we arrive at the final answer
${\displaystyle f^{-1}(x)={\frac {2x+1}{3-4x}}}$
${\displaystyle f^{-1}(x)={\frac {2x+1}{3-4x}}}$