Let's solve the last question, which asks us to find the composite function and select the correct choice.

We are given two functions:

• $f(x) = x^2$
• $g(x) = 20 + x$

We need to evaluate $f(g(x))$ and $g(f(x))$, which represent the compositions of functions.

1. $f \circ g(x) = f(g(x))$

To find $f(g(x))$, substitute $g(x) = 20 + x$ into $f(x) = x^2$: $f(g(x)) = f(20 + x) = (20 + x)^2$ So, $f(g(x)) = (20 + x)^2$.

Thus, the correct answer for $f(g(x))$ is: $B. \, y = f(g(x)) = (20 + x)^2$

2. $g \circ f(x) = g(f(x))$

To find $g(f(x))$, substitute $f(x) = x^2$ into $g(x) = 20 + x$: $g(f(x)) = g(x^2) = 20 + x^2$ So, $g(f(x)) = 20 + x^2$.

The correct answer for $g(f(x))$ is: $A. \, y = g(f(x)) = 20 + x^2$

Both the answers provided are correct as per your choices:

• $f(g(x)) = (20 + x)^2$
• $g(f(x)) = 20 + x^2$

Let me know if you'd like further details or clarification! Here are five questions related to this topic:

1. What is the difference between $f \circ g(x)$ and $g \circ f(x)$?
2. How do you prove the properties of composite functions?
3. Can the order of composition change the result?
4. What are some real-world applications of composite functions?
5. How do composite functions relate to inverse functions?

Tip: When composing functions, always start by substituting the inside function into the outside function.