Let's first break down the given functions:

• f(s) represents the perimeter of a regular pentagon with side length $s$ cm.
• g(p) represents the side length of a regular pentagon given the perimeter $p$ cm.
• h(s) represents the area of a regular pentagon given side length $s$ cm.
• j(a) represents the side length of a regular pentagon given the area $a$ cm².

Part 1: Use function notation for a regular pentagon whose perimeter is 132 cm.

We know that the perimeter $p$ of a regular pentagon is given by: $p = 5s$ where $s$ is the side length.

Given the perimeter is 132 cm, we want to find the side length $s$. Using the function $g(p)$ which gives the side length given the perimeter, we can write: $g(132)$ This represents the side length of the regular pentagon whose perimeter is 132 cm.

Next, we want the area $a$ of the pentagon. Using the function $h(s)$ which gives the area of the pentagon given the side length, we can express the area as: $h(g(132))$ This represents the area of the regular pentagon whose perimeter is 132 cm.

Part 2: Use function notation for the perimeter of a regular pentagon whose area is 6.08 cm².

We are given the area $a = 6.08$ cm². We want to find the side length $s$ of the pentagon. Using the function $j(a)$ which gives the side length given the area, we can write: $j(6.08)$ This represents the side length of the regular pentagon whose area is 6.08 cm².

Finally, we want to find the perimeter $p$. Since the perimeter is related to the side length by $p = 5s$, we use the function $f(s)$ to get the perimeter: $f(j(6.08))$ This represents the perimeter of the regular pentagon whose area is 6.08 cm².

Part 3: Determine which functions make sense in context.

We need to evaluate each of the following:

1. $g(f(s))$: This means first finding the perimeter $f(s)$ for a given side length $s$, and then finding the side length $g(p)$ given that perimeter. This is valid because it’s essentially calculating the side length given the perimeter, so this function makes sense.

2. $j(h(s))$: This means first finding the area $h(s)$ for a given side length $s$, and then using that area to find the side length $j(a)$. This also makes sense, as you are correctly reversing the process from area to side length.

3. $f(h(s))$: This means first finding the area $h(s)$ for a given side length $s$, and then trying to find the perimeter $f(s)$ from the area. This does not make sense because the perimeter depends on the side length, not the area, and $f$ requires the side length, not the area.

4. $f(j(a))$: This means first finding the side length $j(a)$ given the area $a$, and then using that side length to find the perimeter $f(s)$. This makes sense, as it correctly calculates the perimeter from the side length obtained from the area.

Final Answer:

The valid functions in context are:

• $g(f(s))$
• $j(h(s))$
• $f(j(a))$

Would you like further clarification on any part of this? Or have more questions on similar concepts? Here are some questions to explore further:

1. How would you calculate the perimeter given the area of a hexagon instead of a pentagon?
2. How would you derive the formula for the area of a regular polygon with $n$ sides?
3. What would happen if you used the inverse functions for different shapes, such as a square?
4. How do you compute the interior angle of a regular polygon based on its side length?
5. How do the functions change if the polygon is irregular?

Tip: Always ensure the input to each function makes sense in terms of the domain and context, especially when working with geometric functions!