https://file.aiquickdraw.com/mathbot/file/1726716962793g32ixkg4.jpg

Let's work through the three problems provided in the image. The goal is to find:

1. \((f \circ g)(x)\), which means \(f(g(x))\),
2. \((g \circ f)(x)\), which means \(g(f(x))\), and
3. \((f \circ g)(2)\), which means applying \(f(g(2))\).

### 1. \( f(x) = 2x \), \( g(x) = x + 5 \)
- \((f \circ g)(x)\): This means \(f(g(x)) = f(x + 5)\).
  \[
  f(x + 5) = 2(x + 5) = 2x + 10
  \]
  So, \((f \circ g)(x) = 2x + 10\).

- \((g \circ f)(x)\): This means \(g(f(x)) = g(2x)\).
  \[
  g(2x) = 2x + 5
  \]
  So, \((g \circ f)(x) = 2x + 5\).

- \((f \circ g)(2)\): First, find \(g(2)\), then apply \(f\).
  \[
  g(2) = 2 + 5 = 7
  \]
  Now, find \(f(7)\):
  \[
  f(7) = 2(7) = 14
  \]
  So, \((f \circ g)(2) = 14\).

---

### 2. \( f(x) = 4x \), \( g(x) = x - 3 \)
- \((f \circ g)(x)\): This means \(f(g(x)) = f(x - 3)\).
  \[
  f(x - 3) = 4(x - 3) = 4x - 12
  \]
  So, \((f \circ g)(x) = 4x - 12\).

- \((g \circ f)(x)\): This means \(g(f(x)) = g(4x)\).
  \[
  g(4x) = 4x - 3
  \]
  So, \((g \circ f)(x) = 4x - 3\).

- \((f \circ g)(2)\): First, find \(g(2)\), then apply \(f\).
  \[
  g(2) = 2 - 3 = -1
  \]
  Now, find \(f(-1)\):
  \[
  f(-1) = 4(-1) = -4
  \]
  So, \((f \circ g)(2) = -4\).

---

### 3. \( f(x) = 2x + 1 \), \( g(x) = 3x - 2 \)
- \((f \circ g)(x)\): This means \(f(g(x)) = f(3x - 2)\).
  \[
  f(3x - 2) = 2(3x - 2) + 1 = 6x - 4 + 1 = 6x - 3
  \]
  So, \((f \circ g)(x) = 6x - 3\).

- \((g \circ f)(x)\): This means \(g(f(x)) = g(2x + 1)\).
  \[
  g(2x + 1) = 3(2x + 1) - 2 = 6x + 3 - 2 = 6x + 1
  \]
  So, \((g \circ f)(x) = 6x + 1\).

- \((f \circ g)(2)\): First, find \(g(2)\), then apply \(f\).
  \[
  g(2) = 3(2) - 2 = 6 - 2 = 4
  \]
  Now, find \(f(4)\):
  \[
  f(4) = 2(4) + 1 = 8 + 1 = 9
  \]
  So, \((f \circ g)(2) = 9\).

---

Let me know if you'd like more details or if you have any further questions!

Here are 5 questions that could expand your understanding:
1. How do the compositions of functions differ when \(f(x)\) and \(g(x)\) are nonlinear?
2. What happens to the composition if one of the functions is a constant?
3. How would you approach finding \((f \circ g)(x)\) if \(f(x)\) and \(g(x)\) involve fractions?
4. How can compositions of functions be applied in real-world scenarios?
5. Can you derive the inverse of a composed function \((f \circ g)^{-1}(x)\)?

**Tip:** When composing functions, always start from the innermost function and work outward step by step to avoid mistakes.

Find (f ∘ g)(x), (g ∘ f)(x), and (f ∘ g)(2) for the following functions: 1) f(x) = 2x, g(x) = x + 5 2) f(x) = 4x, g(x) = x - 3 3) f(x) = 2x + 1, g(x) = 3x - 2

Solve function compositions for f(x) and g(x) with step-by-step solutions. Learn how to find (f ∘ g)(x), (g ∘ f)(x), and (f ∘ g)(2) for linear functions, covering essential algebraic concepts.

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