To find the inverse function $f^{-1}(x)$ for the given function $f(x) = e^{4x} - 3$, follow these steps:

1. Start with the function:
   $y = f(x) = e^{4x} - 3$

2. Swap $x$ and $y$ to solve for the inverse function:
   $x = e^{4y} - 3$

3. Isolate the exponential term:
   $x + 3 = e^{4y}$

4. Take the natural logarithm (ln) of both sides to solve for $y$:
   $\ln(x + 3) = 4y$

5. Solve for $y$:
   $y = \frac{1}{4} \ln(x + 3)$

So, the inverse function is: $f^{-1}(x) = \frac{1}{4} \ln(x + 3)$

Would you like more details or have any questions?

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Related Questions:

1. What is the domain of $f^{-1}(x)$?
2. How do you find the range of $f^{-1}(x)$?
3. What happens if the function $f(x)$ is shifted vertically?
4. How do you differentiate the inverse function $f^{-1}(x)$?
5. Can the process used to find $f^{-1}(x)$ be applied to other types of functions?

Tip: When finding an inverse function, always check that the original function is one-to-one by ensuring that it passes the Horizontal Line Test.