https://file.aiquickdraw.com/mathbot/file/1726919718254d1wd04vv.jpg

The image contains two functions \( f(x) \) and \( g(x) \) defined piecewise, and the task is to compute \( (f \circ g)(x) \), which is the composition of \( f(x) \) and \( g(x) \).

### Functions Definition:
- \( f(x) = \begin{cases} 
x + 1, & x \leq 1 \\
2x + 1, & 1 < x \leq 2 
\end{cases} \)
  
- \( g(x) = \begin{cases} 
x^2, & -1 \leq x < 2 \\
x + 2, & 2 \leq x \leq 3 
\end{cases} \)

The goal is to find \( (f \circ g)(x) = f(g(x)) \).

### Step-by-step:
1. **For \( -1 \leq x < 2 \):**
   - \( g(x) = x^2 \), so we need to find \( f(g(x)) = f(x^2) \).
     - If \( 0 \leq x < 1 \), then \( g(x) = x^2 \leq 1 \), so \( f(g(x)) = f(x^2) = x^2 + 1 \).
     - If \( 1 < x < \sqrt{2} \), then \( g(x) = x^2 > 1 \), so \( f(g(x)) = f(x^2) = 2x^2 + 1 \).

2. **For \( 2 \leq x \leq 3 \):**
   - \( g(x) = x + 2 \), and since \( 2 \leq g(x) \leq 5 \), we are in the domain where \( f(x) = 2x + 1 \), hence \( f(g(x)) = f(x + 2) = 2(x + 2) + 1 = 2x + 5 \).

This matches part of the answers in the image. Would you like a more detailed explanation for the specific parts or have questions?

Here are 5 related questions:
1. How do we determine the domain for the composed function \( (f \circ g)(x) \)?
2. What happens if the range of \( g(x) \) is outside the domain of \( f(x) \)?
3. How would the composition change if \( g(x) \) had a different piecewise definition?
4. What is the significance of calculating \( (f \circ g)(x) \) in real-world problems?
5. Can you find \( (g \circ f)(x) \) using the same approach?

**Tip:** Always check the domain of each function carefully when composing piecewise functions!

Diberikan fungsi f(x) dan g(x) yang terdefinisi secara potongan, temukan komposisi (f ∘ g)(x).

Learn how to find the composition of two piecewise functions, f(x) and g(x). The step-by-step solution covers key concepts in function composition and piecewise definitions, suitable for high school students.

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