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.

Discover how to find the inverse of the exponential function f^-1(x) = e^2x+5, explained with step-by-step reasoning. Learn about key concepts like logarithms, exponentials, and inverse functions. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation