004 Sample Final A, Problem 1

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

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

Solution:

Step 1:
We start by replacing $f(x)$ with $y$.
This leaves us with $y={\frac {3x-1}{4x+2}}$

Step 2:
Now, we swap $x$ and $y$ to get $x={\frac {3y-1}{4y+2}}$.

Step 3:
Starting with $x={\frac {3y-1}{4y+2}}$, we multiply both sides by $4y+2$ to get
$x(4y+2)=3y-1$.
Now, we need to get all the $y$ terms on one side. So, adding 1 and $-4xy$ to both sides we get
$2x+1=3y-4xy$.

Step 4:
Factoring out $y$, we get $2x+1=y(3-4x)$. Now, dividing by $(3-4x)$, we get
${\frac {2x+1}{3-4x}}=y$. Replacing $y$ with $f^{-1}(x)$, we arrive at the final answer
$f^{-1}(x)={\frac {2x+1}{3-4x}}$

Final Answer:
$f^{-1}(x)={\frac {2x+1}{3-4x}}$

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