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.