To find the inverse function $f^{-1}(x)$ of the given function $f(x+1) = 2x$, we need to express $x$ in terms of $y$, where $y = f(x+1)$.

Step 1: Start with the original function

The function is given as: $f(x+1) = 2x$ Let's set $y = f(x+1)$, so: $y = 2x$

Step 2: Solve for $x$

To find the inverse, solve for $x$ in terms of $y$: $x = \frac{y}{2}$

Step 3: Express the inverse function

Remember, $y = f(x+1)$, and we found that $x = \frac{y}{2}$. Since $y = f(x+1)$, replace $y$ with $x$ in the inverse function: $f^{-1}(x) = \frac{x}{2} - 1$

Final Answer

The inverse function $f^{-1}(x)$ is: $\boxed{f^{-1}(x) = \frac{x}{2} - 1}$

This is the function that, when applied to the output of $f(x+1)$, returns the original input.

Do you have any questions or would you like further details?

Here are some related questions to explore:

1. How can you verify if a function and its inverse are correct?
2. What are the conditions under which a function has an inverse?
3. How do you find the inverse of a more complex function, such as a quadratic function?
4. What is the graphical relationship between a function and its inverse?
5. Can you find the inverse of the function $g(3x) = x + 6$?

Tip: To verify the correctness of an inverse function, compose the original function with its inverse (i.e., $f(f^{-1}(x))$ and $f^{-1}(f(x))$). Both compositions should return $x$.