https://file.aiquickdraw.com/mathbot/file/17266708598237rm2ug8u.jpg

The image shows a mathematical problem in Indonesian that translates to: 

"If \( f(x) = 2x - 6 \), then \( f^{-1}(x) = \) ..."

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

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

2. Swap \( x \) and \( y \) to find the inverse function:  
   \[
   x = 2y - 6
   \]

3. Solve for \( y \):  
   \[
   x + 6 = 2y
   \]  
   \[
   y = \frac{x + 6}{2}
   \]

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

So, the inverse function is \( f^{-1}(x) = \frac{x + 6}{2} \).

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How do you find the inverse of a linear function in general?
2. What are the graphical interpretations of a function and its inverse?
3. What is the domain and range of \( f(x) = 2x - 6 \) and \( f^{-1}(x) \)?
4. What happens if you compose a function with its inverse, \( f(f^{-1}(x)) \)?
5. How does finding the inverse change if the function is not linear?

**Tip:** When finding the inverse of a function, always ensure that the original function is one-to-one, meaning it passes the horizontal line test.

Learn how to find the inverse of the function f(x) = 2x - 6 in a step-by-step process. This problem covers key algebraic concepts related to inverse functions and linear equations. Suitable for students in Grades 10-12.

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