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Insert an algebraic for composite function here. fg=? and Insert an algebraic expression for composite function here. gf=?

We are tasked with finding the composite functions \( f \circ g \) and \( g \circ f \), where the functions \( f(x) = 5x^2 + 2 \) and \( g(x) = 4\sin(x) \) are given.

### Step 1: Finding \( f \circ g \) (i.e., \( f(g(x)) \))
Here, we are asked to substitute \( g(x) = 4\sin(x) \) into the function \( f(x) = 5x^2 + 2 \).

\[
f(g(x)) = f(4\sin(x)) = 5(4\sin(x))^2 + 2
\]
\[
= 5(16\sin^2(x)) + 2 = 80\sin^2(x) + 2
\]

Thus, 
\[
f \circ g = 80\sin^2(x) + 2
\]

### Step 2: Finding \( g \circ f \) (i.e., \( g(f(x)) \))
Now, we substitute \( f(x) = 5x^2 + 2 \) into the function \( g(x) = 4\sin(x) \).

\[
g(f(x)) = g(5x^2 + 2) = 4\sin(5x^2 + 2)
\]

Thus, 
\[
g \circ f = 4\sin(5x^2 + 2)
\]

### Final Results:
- \( f \circ g = 80\sin^2(x) + 2 \)
- \( g \circ f = 4\sin(5x^2 + 2) \)

Would you like more details on how to approach composite functions, or do you have any questions?

Here are 5 related questions for further exploration:
1. How would you compute \( f \circ g \) if \( g(x) = \cos(x) \)?
2. Can composite functions be reversed? What is an inverse function?
3. How do composite functions change if both functions involve trigonometric identities?
4. How do we differentiate \( f \circ g \) and \( g \circ f \)?
5. What happens if we compose more than two functions together?

**Tip:** Always substitute the inner function directly into the outer function, being careful with parentheses to ensure accuracy in calculations.

Insert the composite functions f ∘ g and g ∘ f, when g = 4sin(x) and f = 5x^2 + 2.

Learn how to find the composite functions f ∘ g and g ∘ f for the given equations g(x) = 4sin(x) and f(x) = 5x^2 + 2. This problem involves understanding both trigonometric and quadratic functions and applying them in composite function calculations. Suitable for high school students in Grades 11-12.

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