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

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

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

Step 2: Swap $x$ and $y$ to set up for finding the inverse.

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

Step 3: Solve for $y$.

First, multiply both sides by 5 to eliminate the denominator: $5x = 4y - 3$ Then, add 3 to both sides: $5x + 3 = 4y$ Finally, divide by 4: $y = \frac{5x + 3}{4}$

Step 4: Write the inverse function.

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

So, the inverse of $f(x) = \frac{4x - 3}{5}$ is $f^{-1}(x) = \frac{5x + 3}{4}$.

Would you like more details or clarifications on any steps?

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Here are 5 related questions you might explore:

1. What is the domain and range of the original function $f(x)$?
2. What is the domain and range of the inverse function $f^{-1}(x)$?
3. How do you verify that $f^{-1}(x)$ is indeed the inverse of $f(x)$?
4. Can every function have an inverse? Why or why not?
5. How do transformations affect the inverse of a function?

Tip: When finding the inverse of a function, always check by composing the original and inverse functions to ensure they simplify to $x$.