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!