https://file.aiquickdraw.com/mathbot/file/1727235427352it3wdvu5.jpg

The image shows two functions: 
- \( F(x) = 2x + 1 \)
- \( g(x) = 3x + 6 \)
  
It appears you're asking for the composition of the functions \( g \circ F \), also known as \( g(F(x)) \). To find this, we substitute \( F(x) \) into \( g(x) \).

1. Start with \( g(x) = 3x + 6 \).
2. Replace \( x \) in \( g(x) \) with \( F(x) = 2x + 1 \):
   \[
   g(F(x)) = 3(2x + 1) + 6
   \]
3. Simplify the expression:
   \[
   g(F(x)) = 6x + 3 + 6 = 6x + 9
   \]

So, the composition \( g \circ F(x) = 6x + 9 \).

Would you like more details on this process, or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you find the composition \( F \circ g(x) \)?
2. What is the domain of the composition \( g \circ F \)?
3. How does the composition of functions relate to their inverses?
4. What changes in the composition if the functions are non-linear (e.g., quadratic)?
5. Can you find a scenario where the composition of two functions is the identity function?

**Tip**: When composing functions, always substitute the entire second function into the first, and carefully simplify step-by-step.

Learn how to compute the function composition g(f(x)) where F(x) = 2x + 1 and g(x) = 3x + 6. This guide provides a step-by-step solution to simplify the expression and covers essential concepts like linear functions and function composition. Suitable for Grades 10-12.

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