Let's solve the problem shown in the image step by step.

Problem Breakdown:

1. Given Functions:

   • $g(x) = (2x)^3$
   • $h(x) = \frac{\sqrt{x}}{2}$

2. Task:

   • Find the simplified expression for $g(h(x))$.
   • Determine if $g$ and $h$ are inverses.

Step 1: Compute $g(h(x))$

To find $g(h(x))$, substitute $h(x)$ into $g(x)$: $g(h(x)) = g\left(\frac{\sqrt{x}}{2}\right)$ Now substitute $\frac{\sqrt{x}}{2}$ into $g(x) = (2x)^3$: $g\left(\frac{\sqrt{x}}{2}\right) = \left(2 \cdot \frac{\sqrt{x}}{2}\right)^3 = \left(\sqrt{x}\right)^3 = x^{3/2}$ So, $g(h(x)) = x^{3/2}$.

Step 2: Check if $g$ and $h$ are Inverses

For $g(x)$ and $h(x)$ to be inverses, we need to check if: $g(h(x)) = x \quad \text{and} \quad h(g(x)) = x$

We already found $g(h(x)) = x^{3/2}$, which is not equal to $x$ in general, so they are not inverses.

Final Answers:

1. The simplified expression for $g(h(x))$ is $x^{3/2}$.
2. Functions $g$ and $h$ are not inverses.

Let me know if you have any further questions or need more details!

Related Questions:

1. How would you find the inverse of $g(x)$ if it were invertible?
2. Can you find an example where two functions are inverses of each other?
3. What are the steps to verify if two functions are inverses in general?
4. How would you interpret the result $x^{3/2}$ in terms of transformations?
5. What properties of the functions $g(x)$ and $h(x)$ led to them not being inverses?

Tip:

Always check both compositions $f(g(x))$ and $g(f(x))$ when determining if two functions are inverses.