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.