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.

Learn how to find the inverse function of f(x) = e^4x - 3 with a detailed step-by-step solution. This problem involves exponential functions, inverse functions, and natural logarithms. Suitable for advanced high school students.

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