https://file.aiquickdraw.com/mathbot/file/1726751303053nlyie25y.jpg

In the image, two different function compositions are being addressed. Let's break them down:

### 1. First problem
- Functions:
  - \( f(x) = 2x - 1 \)
  - \( g(x) = x + 3 \)
  
- Task: Find \( (g \circ f)(x) \), which means \( g(f(x)) \).

Step-by-step solution:
1. Start with \( g(f(x)) \).
2. Substituting \( f(x) \) into \( g(x) \), we have:
   \[
   g(f(x)) = g(2x - 1) = (2x - 1) + 3 = 2x + 2
   \]
So, \( (g \circ f)(x) = 2x + 2 \).

There are also specific evaluations such as:
- \( (g \circ f)(-1) = 2(-1) + 2 = 0 \)
- \( (g \circ f)(-3) = 2(-3) + 2 = -4 \)
- \( (g \circ f)(-5) = 2(-5) + 2 = -8 \)

### 2. Second problem
- Functions:
  - \( f(x) = x^2 + 4 \)
  - \( g(x) = 5x + 3 \)

- Task: Find different compositions like \( (f \circ g)(x) \), \( (g \circ f)(x) \), and evaluate them for specific values.

Step-by-step solution:
1. \( (f \circ g)(x) = f(g(x)) \)
   \[
   f(g(x)) = f(5x + 3) = (5x + 3)^2 + 4
   \]
   Expanding \( (5x + 3)^2 \):
   \[
   = 25x^2 + 30x + 9 + 4 = 25x^2 + 30x + 13
   \]
   So, \( (f \circ g)(x) = 25x^2 + 30x + 13 \).

2. \( (g \circ f)(x) = g(f(x)) \)
   \[
   g(f(x)) = g(x^2 + 4) = 5(x^2 + 4) + 3 = 5x^2 + 20 + 3 = 5x^2 + 23
   \]
   So, \( (g \circ f)(x) = 5x^2 + 23 \).

Evaluations:
- \( (f \circ g)(2) = 25(2)^2 + 30(2) + 13 = 25(4) + 60 + 13 = 100 + 60 + 13 = 173 \)
- \( (g \circ f)(6) = 5(6^2) + 23 = 5(36) + 23 = 180 + 23 = 203 \)

---

Would you like more details or explanations on any step? Here are five questions that could help expand your understanding:

1. What is the general process for composing two functions?
2. How do we expand expressions like \( (5x + 3)^2 \) and simplify them?
3. Can we reverse the order of function compositions? What happens if we switch \( f \) and \( g \)?
4. How do we interpret the values when we evaluate compositions like \( (g \circ f)(x) \) at specific points?
5. What are common applications of function compositions in real-world scenarios?

**Tip:** When composing functions, always substitute the entire inner function before simplifying the expression.

Solve two function compositions: f(x) = 2x - 1, g(x) = x + 3 and f(x) = x^2 + 4, g(x) = 5x + 3.

Learn how to solve function compositions like g(f(x)) and f(g(x)) for functions f(x) = 2x - 1, g(x) = x + 3, f(x) = x^2 + 4, and g(x) = 5x + 3. Includes step-by-step evaluations.

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