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In the provided exercises (65–72), you are asked to find two functions \( f(x) \) and \( g(x) \) such that:

\[
(f \circ g)(x) = h(x)
\]
This means \( f(g(x)) = h(x) \), where \( h(x) \) is the given function. Let's go through each exercise to find such \( f \) and \( g \) functions.

### 65. \( h(x) = (2x + 1)^2 \)
We can choose:
- \( g(x) = 2x + 1 \)
- \( f(x) = x^2 \)

This gives:
\[
f(g(x)) = f(2x + 1) = (2x + 1)^2 = h(x)
\]

### 66. \( h(x) = (1 - x)^3 \)
We can choose:
- \( g(x) = 1 - x \)
- \( f(x) = x^3 \)

This gives:
\[
f(g(x)) = f(1 - x) = (1 - x)^3 = h(x)
\]

### 67. \( h(x) = \sqrt[3]{x^2 - 4} \)
We can choose:
- \( g(x) = x^2 - 4 \)
- \( f(x) = \sqrt[3]{x} \)

This gives:
\[
f(g(x)) = f(x^2 - 4) = \sqrt[3]{x^2 - 4} = h(x)
\]

### 68. \( h(x) = \sqrt{9 - x} \)
We can choose:
- \( g(x) = 9 - x \)
- \( f(x) = \sqrt{x} \)

This gives:
\[
f(g(x)) = f(9 - x) = \sqrt{9 - x} = h(x)
\]

### 69. \( h(x) = \frac{1}{x + 2} \)
We can choose:
- \( g(x) = x + 2 \)
- \( f(x) = \frac{1}{x} \)

This gives:
\[
f(g(x)) = f(x + 2) = \frac{1}{x + 2} = h(x)
\]

### 70. \( h(x) = \frac{4}{(5x + 2)^2} \)
We can choose:
- \( g(x) = 5x + 2 \)
- \( f(x) = \frac{4}{x^2} \)

This gives:
\[
f(g(x)) = f(5x + 2) = \frac{4}{(5x + 2)^2} = h(x)
\]

### 71. \( h(x) = (x + 4)^2 + 2(x + 4) \)
We can choose:
- \( g(x) = x + 4 \)
- \( f(x) = x^2 + 2x \)

This gives:
\[
f(g(x)) = f(x + 4) = (x + 4)^2 + 2(x + 4) = h(x)
\]

### 72. \( h(x) = (x + 3)^{3/2} + 4(x + 3)^{1/2} \)
We can choose:
- \( g(x) = x + 3 \)
- \( f(x) = x^{3/2} + 4x^{1/2} \)

This gives:
\[
f(g(x)) = f(x + 3) = (x + 3)^{3/2} + 4(x + 3)^{1/2} = h(x)
\]

---

Would you like any further details on these problems?

Here are 5 related questions for further practice:
1. How do you determine the functions \( f \) and \( g \) in function composition problems?
2. Can you find alternative pairs of \( f \) and \( g \) for each \( h(x) \)?
3. What happens if you reverse \( f \) and \( g \)? Does \( g \circ f \) work for these problems?
4. How do the domains of \( f \), \( g \), and \( h \) affect the choice of functions?
5. Can you write the composition of three functions to represent more complex \( h(x) \)?

**Tip:** When solving composition problems, focus on identifying the "inner" and "outer" parts of the function to split into \( g(x) \) and \( f(x) \) respectively.

Find two functions f and g such that (f ◦ g)(x) = h(x) for exercises 65 to 72.

Learn how to solve function composition problems by finding functions f and g such that (f ◦ g)(x) = h(x). This problem set includes radicals, quadratic functions, and powers of binomials. Suitable for Grades 10-12 students.

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