The problem gives two equations:

1. $f(x) = 2x$
2. $f(g(x)) = 2x - 2$

We are asked to find $g(x)$.

Step 1: Start with the second equation.

We know that: $f(g(x)) = 2x - 2$ Since $f(x) = 2x$, substitute $g(x)$ into the function $f$: $f(g(x)) = 2g(x)$ Therefore: $2g(x) = 2x - 2$

Step 2: Solve for $g(x)$.

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

Thus, $g(x) = x - 1$.

Would you like further clarification or additional details?

Here are 5 related questions:

1. How do you check if a function is one-to-one?
2. What are the conditions for a function to have an inverse?
3. How can you determine the inverse of a linear function?
4. How does composition of functions work in general?
5. What is the domain and range of the function $g(x) = x - 1$?

Tip: When solving problems involving functions, identifying key properties such as linearity can help you manipulate equations more efficiently.