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The problem asks to find \( f(g(x)) \) and \( g(f(x)) \), and to determine if the functions \( f \) and \( g \) are inverses of each other. The functions are given as:

- \( f(x) = 2x \)
- \( g(x) = \frac{x}{2} \)

### Step 1: Find \( f(g(x)) \)
To find \( f(g(x)) \), substitute \( g(x) = \frac{x}{2} \) into the function \( f(x) = 2x \):
\[
f(g(x)) = f\left( \frac{x}{2} \right) = 2 \times \frac{x}{2} = x
\]

### Step 2: Find \( g(f(x)) \)
To find \( g(f(x)) \), substitute \( f(x) = 2x \) into the function \( g(x) = \frac{x}{2} \):
\[
g(f(x)) = g(2x) = \frac{2x}{2} = x
\]

### Conclusion:
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f \) and \( g \) are indeed inverses of each other.

Would you like further details on any part of this solution? 

Here are 5 additional questions to expand your understanding:
1. What are the general conditions for two functions to be inverses?
2. How would you graphically represent inverse functions?
3. How would you verify inverses for more complex functions?
4. Can a function have more than one inverse?
5. What happens if one function fails the test of being an inverse?

**Tip:** When testing for inverse functions, always check if both \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \).

Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.

f(x) = 2x

g(x) = x/2

This problem involves finding the composite functions f(g(x)) and g(f(x)) for f(x) = 2x and g(x) = x/2. Verify if f and g are inverses of each other by checking if f(g(x)) = x and g(f(x)) = x. Suitable for students in Grades 9-12 learning about inverse and composite functions.

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