Find (f ∘ g)(x) and (g ∘ f)(x) for the following functions:
f(x) = 3x + 8 and g(x) = x2 + 1

To find the compositions \((f \circ g)(x)\) and \((g \circ f)(x)\), we need to substitute one function into the other.

Given the functions:
\[ f(x) = 3x + 8 \]
\[ g(x) = x^2 + 1 \]

### Finding \((f \circ g)(x)\)
This represents \(f(g(x))\). We substitute \(g(x)\) into \(f(x)\):

1. First, find \(g(x)\):
   \[ g(x) = x^2 + 1 \]

2. Now, substitute \(g(x)\) into \(f(x)\):
   \[ f(g(x)) = f(x^2 + 1) \]

3. Substitute \(x^2 + 1\) into \(f(x)\):
   \[ f(x^2 + 1) = 3(x^2 + 1) + 8 \]

4. Simplify the expression:
   \[ f(x^2 + 1) = 3x^2 + 3 + 8 \]
   \[ f(x^2 + 1) = 3x^2 + 11 \]

So,
\[ (f \circ g)(x) = 3x^2 + 11 \]

### Finding \((g \circ f)(x)\)
This represents \(g(f(x))\). We substitute \(f(x)\) into \(g(x)\):

1. First, find \(f(x)\):
   \[ f(x) = 3x + 8 \]

2. Now, substitute \(f(x)\) into \(g(x)\):
   \[ g(f(x)) = g(3x + 8) \]

3. Substitute \(3x + 8\) into \(g(x)\):
   \[ g(3x + 8) = (3x + 8)^2 + 1 \]

4. Expand and simplify the expression:
   \[ g(3x + 8) = (3x + 8)(3x + 8) + 1 \]
   \[ g(3x + 8) = 9x^2 + 24x + 24x + 64 + 1 \]
   \[ g(3x + 8) = 9x^2 + 48x + 65 \]

So,
\[ (g \circ f)(x) = 9x^2 + 48x + 65 \]

### Summary
- \((f \circ g)(x) = 3x^2 + 11\)
- \((g \circ f)(x) = 9x^2 + 48x + 65\)

Would you like more details or have any questions? 

Here are 5 questions you can ask next:
1. How do you find the inverse of a function?
2. What is the derivative of a function and how is it calculated?
3. Can you explain the concept of limits in calculus?
4. How do you solve quadratic equations?
5. What are the properties of logarithmic functions?

**Tip:** When composing functions, always ensure to carefully substitute the inner function into the outer function and simplify the result step-by-step.

Learn how to find the compositions (f ∘ g)(x) and (g ∘ f)(x) for the functions f(x) = 3x + 8 and g(x) = x^2 + 1. This problem involves function composition and quadratic functions, demonstrating step-by-step calculations. Suitable for high school students studying algebra and function theory.

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