To find the inverse of the function $f(x) = e^{5x} - 1$, follow these steps:

Step 1: Replace $f(x)$ with $y$

We start by writing the function as: $y = e^{5x} - 1$

Step 2: Solve for $x$

Now, we need to isolate $x$ in terms of $y$. First, add 1 to both sides: $y + 1 = e^{5x}$ Next, take the natural logarithm (ln) of both sides to remove the exponential: $\ln(y + 1) = 5x$ Now, solve for $x$ by dividing both sides by 5: $x = \frac{1}{5} \ln(y + 1)$

Step 3: Replace $y$ with $x$

Since we're finding the inverse, we replace $y$ with $x$, giving the inverse function: $f^{-1}(x) = \frac{1}{5} \ln(x + 1)$

Thus, the inverse of the function $f(x) = e^{5x} - 1$ is: $f^{-1}(x) = \frac{1}{5} \ln(x + 1)$

Do you want more details or have any questions? Here are 5 related questions for practice:

1. What is the domain of the inverse function $f^{-1}(x)$?
2. How do you find the inverse of a logarithmic function?
3. What is the range of $f(x) = e^{5x} - 1$?
4. Can you find the derivative of $f^{-1}(x)$?
5. What happens to the graph of a function when you take its inverse?

Tip: Always check that the function is one-to-one before finding its inverse, as not all functions have inverses.