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. **Take the natural logarithm (ln) of both sides to solve for \( y \):**  
   \[
   \ln(x) = 4y - 3
   \]

4. **Isolate \( y \):**  
   \[
   4y = \ln(x) + 3
   \]
   \[
   y = \frac{\ln(x) + 3}{4}
   \]

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

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 determine if \( f(x) \) is one-to-one?
3. What is the range of the original function \( f(x) \)?
4. How do you differentiate \( f^{-1}(x) \)?
5. Can this method be used for functions with different exponents, such as \( f(x) = e^{2x-5} \)?

**Tip:** Always remember that the domain of the inverse function is the range of the original function, and vice versa.

Learn how to find the inverse function of f(x) = e^(4x-3) using step-by-step calculations. This problem involves exponential functions and natural logarithms, suitable for advanced high school and above.

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