To find the inverse of the given function:

$f(x) = \frac{3x + 1}{-5x + 2}$

We proceed as follows:

Step 1: Replace $f(x)$ with $y$

$y = \frac{3x + 1}{-5x + 2}$

Step 2: Swap $x$ and $y$

To find the inverse, switch $x$ and $y$: $x = \frac{3y + 1}{-5y + 2}$

Step 3: Solve for $y$

Multiply both sides by $-5y + 2$ to eliminate the denominator: $x(-5y + 2) = 3y + 1$

Distribute $x$ on the left-hand side: $-5xy + 2x = 3y + 1$

Rearrange terms to isolate $y$ terms on one side: $-5xy - 3y = -2x + 1$

Factor out $y$ on the left-hand side: $y(-5x - 3) = -2x + 1$

Solve for $y$: $y = \frac{-2x + 1}{-5x - 3}$

Step 4: Write the inverse

The inverse function is: $f^{-1}(x) = \frac{-2x + 1}{-5x - 3}$

Final Answer:

$f^{-1}(x) = \frac{-2x + 1}{-5x - 3}$

Would you like me to verify this inverse or clarify any steps?