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 $f$).