To find the inverse of the function $J(x) = 4x - 1$, we follow these steps:

1. Replace $J(x)$ with $y$: $y = 4x - 1$

2. Swap $x$ and $y$ to find the inverse: $x = 4y - 1$

3. Solve for $y$ in terms of $x$:

   $x + 1 = 4y$ $y = \frac{x + 1}{4}$

4. Replace $y$ with $J^{-1}(x)$: $J^{-1}(x) = \frac{x + 1}{4}$

So, the inverse of the function $J(x) = 4x - 1$ is: $J^{-1}(x) = \frac{x + 1}{4}$

Would you like more details, or do you have any questions?

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Here are five related questions you might find interesting:

1. How do you verify if two functions are inverses of each other?
2. What is the inverse of $J(x) = 4x + 3$?
3. How do you find the inverse of a quadratic function?
4. What happens to the graph of a function when you find its inverse?
5. How do you interpret the inverse of a function in real-world contexts?

Tip: When finding the inverse of a function, always check if the function is one-to-one, meaning that each $y$-value corresponds to exactly one $x$-value. This ensures that the inverse is also a function.