The given function is:

$f^{-1}(x) = e^{2x + 5}$

This represents the inverse function of some original function $f(x)$. To find $f(x)$, we need to follow these steps:

Step 1: Replace $f^{-1}(x)$ with $y$

Let $y = f^{-1}(x) = e^{2x + 5}$.

Step 2: Swap $x$ and $y$

We switch $x$ and $y$ to find the original function: $x = e^{2y + 5}.$

Step 3: Solve for $y$

To solve for $y$, follow these steps:

1. Take the natural logarithm on both sides to eliminate the exponential function: $\ln(x) = 2y + 5.$
2. Isolate $y$: $2y = \ln(x) - 5,$ $y = \frac{\ln(x) - 5}{2}.$

Thus, the original function is: $f(x) = \frac{\ln(x) - 5}{2}.$

Summary:

The original function is: $f(x) = \frac{\ln(x) - 5}{2}.$

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you find the derivative of $f(x) = \frac{\ln(x) - 5}{2}$?
2. What are the domain and range of $f(x)$?
3. How do you compute the inverse of logarithmic functions in general?
4. How can we verify that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$?
5. How does the natural logarithm function relate to exponential functions?

Tip: When working with logarithmic and exponential equations, taking logarithms on both sides is a common strategy to simplify the expression.