To find the inverse of the function $f(x) = \frac{-3x - 4}{-5x + 1}$, follow these steps:

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

$y = \frac{-3x - 4}{-5x + 1}$

Step 2: Swap $x$ and $y$:

$x = \frac{-3y - 4}{-5y + 1}$

Step 3: Solve for $y$:

Multiply both sides by $-5y + 1$ to eliminate the denominator: $x(-5y + 1) = -3y - 4$

Expand: $-5xy + x = -3y - 4$

Rearrange terms to isolate $y$: $-5xy + 3y = -x - 4$

Factor out $y$ on the left-hand side: $y(-5x + 3) = -x - 4$

Solve for $y$: $y = \frac{-x - 4}{-5x + 3}$

Step 4: Write the inverse:

$f^{-1}(x) = \frac{-x - 4}{-5x + 3}$

Final Answer:

$f^{-1}(x) = \frac{-x - 4}{-5x + 3}$

Would you like further details or have any questions?

Related Questions:

1. How do you verify if two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. Can this function's inverse be graphed? If so, how does it look compared to the original?
4. How do asymptotes behave in the graph of the inverse function?
5. What happens to the domain and range when finding the inverse?

Tip:

Always check if the original function is one-to-one (passes the horizontal line test) before finding its inverse.