https://file.aiquickdraw.com/mathbot/file/1730148584657a0hqefs3.jpg

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = 2x + 6 \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = 2x + 6
   \]

2. **Swap \( x \) and \( y \) to solve for the inverse:**
   \[
   x = 2y + 6
   \]

3. **Isolate \( y \):**
   Subtract 6 from both sides:
   \[
   x - 6 = 2y
   \]
   Then, divide both sides by 2:
   \[
   y = \frac{x - 6}{2}
   \]

4. **Replace \( y \) with \( f^{-1}(x) \):**
   \[
   f^{-1}(x) = \frac{x - 6}{2}
   \]

So, in the format of the answer on the screen, the solution is:
\[
f^{-1}(x) = \frac{1}{2}x - 3
\]

### Filling in the Blanks:
1. The coefficient of \( x \) in \( f^{-1}(x) \) is \( \frac{1}{2} \).
2. The constant term is \( -3 \).

Would you like further details on any part of this solution?

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### Related Questions:
1. How do you find the inverse of functions with a different slope and intercept?
2. What are some key properties of inverse functions?
3. How can you verify that two functions are inverses of each other?
4. How does the concept of inverse functions apply to real-world scenarios?
5. What is the significance of finding the inverse of a function in calculus?

### Tip:
To verify that \( f^{-1}(x) \) is correct, compose \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \); if both equal \( x \), then the functions are true inverses.

Learn how to find the inverse of the linear function f(x) = 2x + 6. This step-by-step solution covers essential algebraic concepts like inverse functions and isolating variables. Ideal for students in Grades 9-10.

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