F ( X ) = X +3 G ( x ) = 2 X + 7 H ( X ) = X square how to do this using fo ( goh) oh (fog)



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

---

### 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, \( G(F(H(X))) = 2X^2 + 13 \).

---

### Final Results:

- \( F(G(H(X))) = 2X^2 + 10 \)
- \( G(F(H(X))) = 2X^2 + 13 \)

---

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!

F(X) = X + 3, G(X) = 2X + 7, H(X) = X^2. How to do this using F ∘ (G ∘ H) and G ∘ (F ∘ H)?

Learn how to solve the function compositions F(G(H(X))) and G(F(H(X))) with given functions F(X) = X + 3, G(X) = 2X + 7, and H(X) = X^2. This problem explores key algebraic concepts, including function composition and quadratic functions. Suitable for high school students in Grades 9-12.

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