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.