https://file.aiquickdraw.com/mathbot/file/1727464010004v1rybuza.jpg

Let's break down the two problems from the image involving function compositions \( f \circ g \) and \( g \circ f \).

### Problem 1:
Given:
\[
f(x) = 4x + 7 \quad \text{and} \quad g(x) = x^8
\]

1. **\( f \circ g \) (which means \( f(g(x)) \)):**
   - First, substitute \( g(x) = x^8 \) into \( f(x) = 4x + 7 \).
   - \( f(g(x)) = f(x^8) = 4(x^8) + 7 = 4x^8 + 7 \).

2. **\( g \circ f \) (which means \( g(f(x)) \)):**
   - Now, substitute \( f(x) = 4x + 7 \) into \( g(x) = x^8 \).
   - \( g(f(x)) = g(4x + 7) = (4x + 7)^8 \).

### Problem 2:
Given:
\[
f(x) = \sqrt{x^2 + 7} \quad \text{and} \quad g(x) = x^2 + 2
\]

1. **\( f \circ g \) (which means \( f(g(x)) \)):**
   - Substitute \( g(x) = x^2 + 2 \) into \( f(x) = \sqrt{x^2 + 7} \).
   - \( f(g(x)) = f(x^2 + 2) = \sqrt{(x^2 + 2)^2 + 7} = \sqrt{x^4 + 4x^2 + 4 + 7} = \sqrt{x^4 + 4x^2 + 11} \).

2. **\( g \circ f \) (which means \( g(f(x)) \)):**
   - Substitute \( f(x) = \sqrt{x^2 + 7} \) into \( g(x) = x^2 + 2 \).
   - \( g(f(x)) = g(\sqrt{x^2 + 7}) = (\sqrt{x^2 + 7})^2 + 2 = x^2 + 7 + 2 = x^2 + 9 \).

### Summary of Solutions:
- For Problem 1:
  - \( f \circ g = 4x^8 + 7 \)
  - \( g \circ f = (4x + 7)^8 \)

- For Problem 2:
  - \( f \circ g = \sqrt{x^4 + 4x^2 + 11} \)
  - \( g \circ f = x^2 + 9 \)

Would you like more details on any of these steps? Here are five related questions for further exploration:

1. How does the composition of functions relate to transformations in real-world contexts?
2. What are the properties of function composition in terms of commutativity and associativity?
3. How can we differentiate or integrate composed functions like \( f \circ g \) and \( g \circ f \)?
4. What are the common errors when dealing with function compositions in algebra?
5. How would the result change if the functions \( f(x) \) and \( g(x) \) were non-polynomial?

**Tip:** Always evaluate the inner function first in function compositions (i.e., \( f(g(x)) \) means apply \( g(x) \) first, then \( f \)).

f(x) = 4x + 7 and g(x) = x^8; f(x) = sqrt(x^2 + 7) and g(x) = x^2 + 2. Find f ∘ g and g ∘ f for both cases.

Explore the compositions of functions f(x) = 4x + 7 and g(x) = x^8, as well as f(x) = sqrt(x^2 + 7) and g(x) = x^2 + 2. This example illustrates how to compute f ∘ g and g ∘ f using polynomials and radical functions. Suitable for students in Grades 9-12.

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