Let's solve the problem given in the image.

We have: $f(x) = 2x$ and $f(g(x)) = 2x - 2$

We need to find the function $g(x)$.

Step 1: Substitute $g(x)$ into $f(x)$

Since $f(g(x)) = 2x - 2$, and we know that $f(x) = 2x$, we substitute $g(x)$ for $x$ in the function $f(x)$: $f(g(x)) = 2g(x)$ Thus, we have: $2g(x) = 2x - 2$

Step 2: Solve for $g(x)$

To isolate $g(x)$, divide both sides of the equation by 2: $g(x) = x - 1$

So the function $g(x)$ is: $g(x) = x - 1$

This is the required function.

Would you like more details or have any questions about this solution? Here are some related questions to consider:

1. What happens if we change $f(x)$ to a different linear function?
2. How do we verify that $g(x) = x - 1$ satisfies the original equation?
3. What would $g(x)$ be if $f(x) = 3x$ instead of $2x$?
4. Can we use this method for non-linear functions $f(x)$?
5. How would this problem change if we were given $f(g(x)) = ax + b$?

Tip: Always verify your solution by substituting it back into the original equation to ensure it satisfies all given conditions.