https://file.aiquickdraw.com/mathbot/file/17260537956933ongpsei.jpg

Let's go through the mathematical problems shown in the image one by one. 

1. **Buatlah grafik dari masing-masing fungsi berikut:**
   - \( f(x) = -x^2 + 1 \)
   - \( f(x) = -x^2 - 2x + 2 \)

2. **Tentukan domain dan range dari fungsi \( f(x) = -x^2 - 1 \) dan gambarkan grafik fungsinya.**

3. **Diketahui \( f(x) = |x| + 1 \) dan \( g(x) = \sqrt{x - \sqrt{x}} \):**
   - a. Tentukan \( D_f, R_f, D_g, \) dan \( R_g \).
   - b. Selidiki apakah \( (f \circ g)(x) \) dan \( (g \circ f)(x) \) ada?
   - c. Tentukan \( D_{f \circ g}, R_{f \circ g}, D_{g \circ f}, \) dan \( R_{g \circ f} \).

4. **Diketahui \( f(x) = \sqrt{1 + x} \) dan \( g(x) = -x^2 - 1 \):**
   - a. Tentukan \( D_{f \circ g} \) dan \( R_{f \circ g} \).
   - b. Tentukan \( D_{g \circ f} \) dan \( R_{g \circ f} \).

Let's solve them step by step:

### Problem 1:
#### a. \( f(x) = -x^2 + 1 \)
- This is a parabola opening downwards with its vertex at \((0, 1)\).

#### b. \( f(x) = -x^2 - 2x + 2 \)
- This is a parabola opening downwards. First, complete the square to find its vertex:
  \[
  f(x) = -x^2 - 2x + 2 = -(x^2 + 2x - 2) = -(x+1)^2 + 3
  \]
- The vertex is at \((-1, 3)\).

### Problem 2:
#### \( f(x) = -x^2 - 1 \)
- This is a parabola opening downwards with its vertex at \((0, -1)\).
- **Domain**: All real numbers, \(\mathbb{R}\).
- **Range**: All \( y \leq -1 \).

### Problem 3:
#### Given \( f(x) = |x| + 1 \) and \( g(x) = \sqrt{x - \sqrt{x}} \):
- \( f(x) = |x| + 1 \):
  - **Domain**: All real numbers, \(\mathbb{R}\).
  - **Range**: \( y \geq 1 \).
- \( g(x) = \sqrt{x - \sqrt{x}} \):
  - **Domain**: \( x \geq 0 \) because of the square root.
  - For \( x - \sqrt{x} \geq 0 \), let \( u = \sqrt{x} \). Then \( u^2 - u \geq 0 \) implies \( u(u - 1) \geq 0 \). So \( u \leq 0 \) or \( u \geq 1 \).
  - Therefore, \( x = u^2 \) means \( x \geq 1 \).
  - **Range**: \( y \geq 0 \).
- Investigate compositions \( (f \circ g)(x) \) and \( (g \circ f)(x) \):
  - \( (f \circ g)(x) = f(g(x)) = f(\sqrt{x - \sqrt{x}}) \).
  - \( (g \circ f)(x) = g(f(x)) = g(|x| + 1) \).

### Problem 4:
#### Given \( f(x) = \sqrt{1 + x} \) and \( g(x) = -x^2 - 1 \):
- \( f(x) = \sqrt{1 + x} \):
  - **Domain**: \( x \geq -1 \).
  - **Range**: \( y \geq 0 \).
- \( g(x) = -x^2 - 1 \):
  - **Domain**: All real numbers, \(\mathbb{R}\).
  - **Range**: All \( y \leq -1 \).

If you want any detailed graphs, let me know. Do you have any questions or need more details about any specific part of the solution? 

Here are 5 related questions you might find interesting:

1. How do you find the vertex of a quadratic function?
2. What is the process of completing the square?
3. How do you determine the domain and range of a composition of functions?
4. How can you graph absolute value functions?
5. What are the steps to analyze and graph a piecewise function?

**Tip:** When finding the domain of a composition of functions, always consider the domain restrictions of both the inner and outer functions.

Buatlah grafik dari fungsi-fungsi berikut, tentukan domain dan range, dan selidiki fungsi komposisi serta invers dari beberapa fungsi yang diberikan.

Learn how to graph quadratic functions, determine the domain and range, and explore the composition of functions, including absolute value and square root functions. This exercise includes detailed examples for high school students (Grades 10-12).

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