https://file.aiquickdraw.com/mathbot/file/1726592751376gtc8om23.jpg

The image contains two functions \( f(x) = 2x \) and \( g(x) = x^2 \). The goal is to find two compositions:

1. \( f \circ g(x) \) 
2. \( g \circ f(x) \)

### 1. \( f \circ g(x) \) (This means \( f(g(x)) \)):
- First, find \( g(x) \). Given \( g(x) = x^2 \), substitute \( g(x) \) into \( f \):
  \[
  f(g(x)) = f(x^2)
  \]
- Now use the definition of \( f(x) = 2x \). Substitute \( x^2 \) for \( x \):
  \[
  f(x^2) = 2(x^2) = 2x^2
  \]
So, \( f \circ g(x) = 2x^2 \).

### 2. \( g \circ f(x) \) (This means \( g(f(x)) \)):
- First, find \( f(x) \). Given \( f(x) = 2x \), substitute \( f(x) \) into \( g \):
  \[
  g(f(x)) = g(2x)
  \]
- Now use the definition of \( g(x) = x^2 \). Substitute \( 2x \) for \( x \):
  \[
  g(2x) = (2x)^2 = 4x^2
  \]
So, \( g \circ f(x) = 4x^2 \).

### Final Results:
- \( f \circ g(x) = 2x^2 \)
- \( g \circ f(x) = 4x^2 \)

Would you like more details on any part of this? Here are some related questions for further exploration:

1. What is the general rule for composing functions?
2. How does the order of composition affect the result?
3. Can you apply these compositions to other functions like trigonometric or logarithmic functions?
4. What happens if you reverse the order of the compositions for different functions?
5. How would you differentiate a composed function like \( f(g(x)) \)?

**Tip**: In function composition, always evaluate the inner function first, then apply the outer function to the result.

f(x) = 2x and g(x) = x^2. Find f∘g(x) and g∘f(x).

Learn how to compose two functions: f(x) = 2x and g(x) = x^2. This problem demonstrates how to calculate f∘g(x) and g∘f(x), showing the steps involved in function composition. Ideal for high school students in grades 9-12.

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