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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.

If f(x) = 2x and f(g(x)) = 2x - 2, then what is g(x)?

Learn how to find g(x) given f(x) = 2x and f(g(x)) = 2x - 2. This problem explores key algebraic concepts, including function composition and inverse functions, and provides a detailed step-by-step solution. Suitable for students in Grades 9-11.

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