The problem presented is a composition of functions, specifically $(f \circ g)(x) = f(g(x))$. Let’s analyze it step by step:

1. Given Functions:

   • $g(x) = \frac{3}{x}$,
   • $f(u) = \frac{u}{2}$, where $u = g(x)$.

2. Composition of Functions:

   • Substituting $g(x)$ into $f(u)$, we get: $f(g(x)) = f\left(\frac{3}{x}\right) = \frac{\frac{3}{x}}{2}.$

3. Simplification:

   • Simplify $f\left(\frac{3}{x}\right)$: $f(g(x)) = \frac{\frac{3}{x}}{2} = \frac{3}{2x}.$

4. Further Operations:

   • The second step in the image shows subtracting 1 from the numerator: $\frac{\frac{3}{x} - 1}{2}.$

   • Combine terms: $\frac{\frac{3 - x}{x}}{2}.$

   • Final simplification gives: $\frac{2x}{3 - x}.$

Thus, $(f \circ g)(x) = \frac{2x}{3 - x}$, which matches the result in the image.

Would you like me to elaborate on any specific step or concept?

Related Questions

1. What are the domain restrictions for $(f \circ g)(x)$ in this case?
2. How does the composition of functions change if $f(u) = u^2$?
3. Can you explain the concept of domain and range in function composition?
4. How would you differentiate $(f \circ g)(x)$ if $f(x)$ and $g(x)$ were both differentiable?
5. What is the significance of $(f \circ g)(x)$ in real-world applications?

Tip: Always check for domain restrictions when dealing with compositions to ensure the result is valid.