https://file.aiquickdraw.com/mathbot/file/1725931133002muvb0x73.jpg

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.

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

In this problem, we solve for g(x) given that f(x) = 2x and f(g(x)) = 2x - 2. The solution involves understanding functions and basic algebraic manipulation. Ideal for high school students learning about functions.

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