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The problem you're working on involves finding the composite function \( (f \circ g)(x) = f(g(x)) \), with functions \( f(x) \) and \( g(x) \) defined piecewise as follows:

### Function \( f(x) \):
\[
f(x) =
\begin{cases}
x + 1, & x \leq 1 \\
2x + 1, & 1 < x \leq 2
\end{cases}
\]

### Function \( g(x) \):
\[
g(x) =
\begin{cases}
x^2, & -1 \leq x < 2 \\
x + 2, & 2 \leq x \leq 3
\end{cases}
\]

### Problem: Find \( (f \circ g)(x) \)

To solve \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \), applying the definition of \( f(x) \) to the values of \( g(x) \) over different intervals of \( x \). Here's a step-by-step breakdown of how we approach this for each interval of \( g(x) \):

### 1. For \( -1 \leq x < 2 \), we have \( g(x) = x^2 \).
- If \( g(x) = x^2 \), apply the definition of \( f(x) \):
  - When \( g(x) \leq 1 \), use \( f(x) = x + 1 \).
  - When \( 1 < g(x) \leq 2 \), use \( f(x) = 2x + 1 \).
  
### 2. For \( 2 \leq x \leq 3 \), we have \( g(x) = x + 2 \).
- If \( g(x) = x + 2 \), apply \( f(g(x)) \) based on the intervals of \( f(x) \).

I can walk through each step in detail and work out the specific form for \( (f \circ g)(x) \) for all intervals. Let me know if you'd like that! 

Do you have any questions or would you like a detailed explanation for each interval?

Here are 5 related questions to consider:
1. How do we handle different intervals in composite functions?
2. What happens when the range of \( g(x) \) crosses the boundaries of \( f(x) \)'s piecewise definition?
3. How do we verify the domain of the composite function?
4. What methods can simplify complex composite functions?
5. How does changing the piecewise definitions of \( f(x) \) and \( g(x) \) affect the overall composition?

### Tip:
Always make sure to clearly define the range of \( g(x) \) and compare it with the intervals in \( f(x) \) to avoid mistakes in composition.

Given the functions f(x) and g(x), where f(x) is defined as f(x) = {x + 1, if x ≤ 1; 2x + 1, if 1 < x ≤ 2} and g(x) is defined as g(x) = {x^2, if -1 ≤ x < 2; x + 2, if 2 ≤ x ≤ 3}, find the composite function (f ∘ g)(x).

Learn how to solve composite functions with piecewise definitions. This problem involves finding (f ∘ g)(x) using the functions f(x) = {x + 1, 2x + 1} and g(x) = {x^2, x + 2} for various intervals of x. Suitable for students in Grades 10-12.

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