To find the inverse function $f^{-1}(x)$ of the given function $f(x) = 1 + 4x$, follow these steps:

Step 1: Express the function as $y = f(x)$

$y = 1 + 4x$

Step 2: Swap $y$ and $x$

To find the inverse, swap $x$ and $y$: $x = 1 + 4y$

Step 3: Solve for $y$

Now, solve this equation for $y$ to find the inverse function: $x - 1 = 4y$ $y = \frac{x - 1}{4}$

Step 4: Express the inverse function

The inverse function is: $f^{-1}(x) = \frac{x - 1}{4}$

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

Would you like further details or have any questions about this solution?

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Here are some related questions you might find interesting:

1. How would you verify that $f^{-1}(x)$ is indeed the inverse of $f(x)$?
2. What is the domain and range of $f(x) = 1 + 4x$?
3. How would the inverse function change if $f(x) = ax + b$ for different values of $a$ and $b$?
4. How can you find the inverse of a quadratic function?
5. If $f(x) = 1 + 4x$, what is the composition $f(f^{-1}(x))$?
6. How does the graph of $f(x)$ compare to the graph of $f^{-1}(x)$?
7. What happens to the inverse function if $f(x)$ is not a one-to-one function?
8. Can every function have an inverse? Why or why not?

Tip: When finding the inverse of a function, ensure that the function is one-to-one. This means each output is associated with exactly one input, making the inverse function well-defined.