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To verify that \( g(x) \) is the inverse of \( f(x) \), we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

Given:
- \( f(x) = 3x \)
- \( g(x) = \frac{1}{3}x \)

Let's apply these functions to verify.

1. \( f(g(x)) = f\left( \frac{1}{3}x \right) = 3 \times \frac{1}{3}x = x \)

2. \( g(f(x)) = g(3x) = \frac{1}{3} \times 3x = x \)

Both conditions hold, so \( g(x) \) is indeed the inverse of \( f(x) \).

Now let's find which option reflects this:

- The second option shows \( \left( \frac{1}{3}x \right)(3x) \), which is equivalent to \( g(f(x)) = x \), confirming the inverse relationship.

Therefore, **the correct answer is the second option.**

Would you like more details or have any questions? Here are some related questions:

1. How do you find the inverse of a function in general?
2. Can a function have more than one inverse?
3. How do graphical interpretations of inverses work?
4. What is the significance of the line \( y = x \) in verifying inverses?
5. How can you test if a function is one-to-one?

**Tip:** Always check both \( f(g(x)) \) and \( g(f(x)) \) to verify inverse relationships.

If f(x) = 3x and g(x) = (1/3)x, which expression could be used to verify that g(x) is the inverse of f(x)?

Explore how to verify that g(x) = (1/3)x is the inverse of f(x) = 3x by applying the concept of inverse functions. Learn about key algebraic principles and how f(g(x)) = x and g(f(x)) = x confirm the inverse relationship. Suitable for Grades 9-12 students.

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