To express the function $h(x) = (3x - 5)^2$ as the composition of two functions $f(g(x))$, we need to find functions $f(x)$ and $g(x)$ such that their composition gives us $h(x)$.

Let's break it down:

1. First, observe that $h(x) = (3x - 5)^2$ can be written in terms of two operations:
   • Step 1: Compute $3x - 5$.
   • Step 2: Square the result of Step 1.

We can assign these steps to different functions:

• Let $g(x) = 3x - 5$ (this represents the inner function).
• Let $f(x) = x^2$ (this represents the outer function, which squares its input).

Now, the composition $f(g(x))$ becomes:

$f(g(x)) = f(3x - 5) = (3x - 5)^2$

Thus, the composition is: $h(x) = f(g(x)) \quad \text{where} \quad f(x) = x^2 \quad \text{and} \quad g(x) = 3x - 5.$

Would you like further details on compositions, or do you have any questions?

Related Questions:

1. How can you compose more than two functions, like $f(g(h(x)))$?
2. What is the general procedure for decomposing functions?
3. How does composition of functions apply in real-world contexts?
4. Can you always decompose any function into two simpler functions?
5. What are the key properties of composite functions?

Tip:

When composing functions, always focus on identifying an "inner" function (what happens first) and an "outer" function (what happens last).