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To find the inverse of the function \( f(x) = 6^x + 2 \), follow these steps:

1. **Step 1: Replace \( f(x) \) with \( y \)**:
   \[
   y = 6^x + 2
   \]
   
2. **Step 2: Swap \( x \) and \( y \)**:
   \[
   x = 6^y + 2
   \]
   
3. **Step 3: Solve for \( y \)**:
   - First, subtract 2 from both sides:
     \[
     x - 2 = 6^y
     \]
   - Now, take the logarithm of both sides. Since the base of the exponent is 6, use the logarithm with base 6:
     \[
     \log_6(x - 2) = y
     \]
     
Thus, the inverse function is:
\[
f^{-1}(x) = \log_6(x - 2)
\]

Would you like more details or have any questions?

Here are some related questions to further your understanding:
1. How do you verify that a function and its inverse are correct?
2. What is the domain and range of \( f(x) = 6^x + 2 \)?
3. What is the relationship between logarithmic and exponential functions?
4. How does the base of the logarithm affect the shape of its graph?
5. How can you find the inverse of more complex exponential functions?

**Tip**: Always check the domain of the inverse function, especially when dealing with logarithms! In this case, \( x - 2 > 0 \), so \( x > 2 \).

Find the equation of the inverse of f(x) = 6^x + 2.

Learn how to find the inverse of the exponential function f(x) = 6^x + 2. This problem involves the use of logarithmic properties to solve for the inverse. Suitable for students in Grades 10-12 studying advanced algebra.

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