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The image contains a mathematical exercise involving function composition. Let's walk through the steps to solve these problems.

### Given Functions:
- \( f(x) = 5x - 3 \)
- \( g(x) = 2x + 3 \)
- \( h(x) = x^2 \)

### 1. **First Problem: Composition of \( f \circ g \) and \( g \circ f \)**

We are asked to find the compositions of \( f(g(x)) \) and \( g(f(x)) \).

- \( (f \circ g)(x) = f(g(x)) \)
  \[
  f(g(x)) = f(2x + 3) = 5(2x + 3) - 3 = 10x + 15 - 3 = 10x + 12
  \]

- \( (g \circ f)(x) = g(f(x)) \)
  \[
  g(f(x)) = g(5x - 3) = 2(5x - 3) + 3 = 10x - 6 + 3 = 10x - 3
  \]

So,
\[
(f \circ g)(x) = 10x + 12, \quad (g \circ f)(x) = 10x - 3
\]

### 2. **Second Problem: Composition of \( (f \circ g) \circ h \) and \( f \circ (g \circ h) \)**

- \( ((f \circ g) \circ h)(x) = (f \circ g)(h(x)) = f(g(h(x))) \)
  \[
  h(x) = x^2, \quad g(h(x)) = g(x^2) = 2x^2 + 3
  \]
  Now, apply \( f \):
  \[
  f(g(h(x))) = f(2x^2 + 3) = 5(2x^2 + 3) - 3 = 10x^2 + 15 - 3 = 10x^2 + 12
  \]

- \( (f \circ (g \circ h))(x) = f(g(h(x))) \) (we already computed this):
  \[
  f(g(h(x))) = 10x^2 + 12
  \]

Thus, both \( (f \circ g) \circ h \) and \( f \circ (g \circ h) \) give the same result, indicating that function composition is associative.

### 3. **Third Problem: Composition with Identity Function \( I(x) = x \)**

We are asked to show that \( f \circ I = I \circ f = f \). Given \( f(x) = 5x - 4 \) and \( I(x) = x \), we compute:

- \( f \circ I(x) = f(I(x)) = f(x) = 5x - 4 \)
- \( I \circ f(x) = I(f(x)) = I(5x - 4) = 5x - 4 \)

Thus, \( f \circ I(x) = I \circ f(x) = f(x) \), meaning the identity function does not change the outcome of the composition.

---

### Summary of Properties:
1. **Non-Commutative**: \( f \circ g \neq g \circ f \)
2. **Associative**: \( (f \circ g) \circ h = f \circ (g \circ h) \)
3. **Identity Function**: \( f \circ I = I \circ f = f \)

Would you like further details or explanation on any part? Here are 5 related questions you can explore:

1. What are other examples of functions that are not commutative?
2. Can function composition ever be commutative?
3. How does function composition relate to matrix multiplication?
4. What happens when we compose a function with itself, \( f \circ f \)?
5. How can function composition be visualized graphically?

**Tip**: When composing functions, always work step-by-step by solving the innermost function first.

Given three functions f(x) = 5x - 3, g(x) = 2x + 3, and h(x) = x^2, perform the compositions (f ∘ g)(x), (g ∘ f)(x), (f ∘ g ∘ h)(x), and analyze their properties: commutativity, associativity, and the identity function.

Explore the composition of functions f(x) = 5x - 3, g(x) = 2x + 3, and h(x) = x^2. This detailed guide covers the properties of function composition, including non-commutativity, associativity, and the role of the identity function. Suitable for Grades 10-12.

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