You are working with three functions:

1. $F(X) = X + 3$
2. $G(X) = 2X + 7$
3. $H(X) = X^2$

Now, you want to find the composition of these functions, specifically:

• $F \circ (G \circ H)$, which means $F(G(H(X)))$
• $G \circ (F \circ H)$, which means $G(F(H(X)))$

Let’s calculate them one by one.

Step 1: Compute $G \circ H$ (i.e., $G(H(X))$)

Start by plugging $H(X) = X^2$ into $G(X)$:

$G(H(X)) = G(X^2) = 2(X^2) + 7 = 2X^2 + 7$

Step 2: Compute $F \circ (G \circ H)$ (i.e., $F(G(H(X)))$)

Now plug $G(H(X)) = 2X^2 + 7$ into $F(X)$:

$F(G(H(X))) = F(2X^2 + 7) = (2X^2 + 7) + 3 = 2X^2 + 10$

So, $F(G(H(X))) = 2X^2 + 10$.

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Step 3: Compute $F \circ H$ (i.e., $F(H(X))$)

Now, plug $H(X) = X^2$ into $F(X)$:

$F(H(X)) = F(X^2) = (X^2) + 3 = X^2 + 3$

Step 4: Compute $G \circ (F \circ H)$ (i.e., $G(F(H(X)))$)

Now, plug $F(H(X)) = X^2 + 3$ into $G(X)$:

$G(F(H(X))) = G(X^2 + 3) = 2(X^2 + 3) + 7 = 2X^2 + 6 + 7 = 2X^2 + 13$

So, $G(F(H(X))) = 2X^2 + 13$.

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Final Results:

• $F(G(H(X))) = 2X^2 + 10$
• $G(F(H(X))) = 2X^2 + 13$

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Would you like more details on any of these steps, or explanations on related concepts?

Here are 5 questions to explore further:

1. How does function composition differ from regular function evaluation?
2. Can compositions be associative, like $(F \circ G) \circ H = F \circ (G \circ H)$?
3. How do domain restrictions affect composite functions?
4. How does the order of function composition affect the result?
5. How can we verify the composite functions graphically?

Tip: Always check the domains of the individual functions when performing compositions to ensure the operations are valid!