F ( X ) = X +3 G ( x ) = 2 X + 7 H ( X ) = X SQUARE HOW TO DO THIS? FIND FOG AND GOF 

To solve the problem of finding the compositions of functions \( F \circ G \) (denoted as \( F(G(x)) \)) and \( G \circ F \) (denoted as \( G(F(x)) \)), we'll use the definitions of the given functions:

- \( F(X) = X + 3 \)
- \( G(X) = 2X + 7 \)
- \( H(X) = X^2 \) (This function is not directly used in finding \( F \circ G \) or \( G \circ F \), but it is part of the given information.)

### 1. Find \( F(G(x)) \):

\( F(G(x)) \) means you substitute \( G(x) \) into \( F(x) \).

Step-by-step:
- Start with \( G(x) = 2x + 7 \).
- Substitute \( G(x) \) into \( F(x) \):

\[
F(G(x)) = F(2x + 7)
\]

Now, use the definition of \( F(x) = x + 3 \):

\[
F(2x + 7) = (2x + 7) + 3
\]

Simplify:

\[
F(G(x)) = 2x + 10
\]

### 2. Find \( G(F(x)) \):

\( G(F(x)) \) means you substitute \( F(x) \) into \( G(x) \).

Step-by-step:
- Start with \( F(x) = x + 3 \).
- Substitute \( F(x) \) into \( G(x) \):

\[
G(F(x)) = G(x + 3)
\]

Now, use the definition of \( G(x) = 2x + 7 \):

\[
G(x + 3) = 2(x + 3) + 7
\]

Simplify:

\[
G(x + 3) = 2x + 6 + 7 = 2x + 13
\]

### Final Results:
- \( F(G(x)) = 2x + 10 \)
- \( G(F(x)) = 2x + 13 \)

Would you like further details or have any questions?

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**Related Questions:**
1. What is the difference between function composition and multiplication?
2. How would the results change if the functions involved were quadratic?
3. Can we find the inverse of \( F \) or \( G \)?
4. How do you graph \( F(G(x)) \) and \( G(F(x)) \)?
5. What happens when you compose more than two functions?

**Tip:** When composing functions, always substitute the entire expression of one function into the other—don’t just substitute the value!

F(X) = X + 3, G(X) = 2X + 7, H(X) = X^2. How to find FOG and GOF?

Learn how to find the compositions of functions F(G(x)) and G(F(x)) with given functions F(X) = X + 3, G(X) = 2X + 7. This step-by-step guide covers function composition concepts suitable for students in Grades 9-12.

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