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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.

Learn how to solve composite functions by evaluating f(g(x)) and g(f(x)) using the given functions f(x) = x^2 and g(x) = 20 + x. This problem covers key algebraic concepts and function composition, suitable for students in Grades 10-12.

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