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Let's go through each question one by one.

### Problem 1
Given:
\[
f(x) = 1 - \sqrt{x - 3}
\]
Determine the domain \( D(f) \) and the range \( R(f) \).

1. **Domain \( D(f) \)**: For \( f(x) = 1 - \sqrt{x - 3} \) to be defined, the expression inside the square root must be non-negative:
   \[
   x - 3 \geq 0 \Rightarrow x \geq 3
   \]
   Therefore, the domain \( D(f) \) is:
   \[
   D(f) = [3, \infty)
   \]

2. **Range \( R(f) \)**: Since \( \sqrt{x - 3} \) yields values from 0 to infinity, \( f(x) = 1 - \sqrt{x - 3} \) will yield values from 1 to negative infinity as \( x \) increases. Thus, the range is:
   \[
   R(f) = (-\infty, 1]
   \]

### Problem 2
Given:
\[
f(x) = \sqrt{x + 1} \quad \text{and} \quad g(x) = x^2 - 1
\]
Determine \( f \circ g \), \( D(f \circ g) \), \( g \circ f \), and \( D(g \circ f) \).

1. **\( f \circ g \)**: The composition \( f \circ g(x) = f(g(x)) \).
   \[
   f(g(x)) = f(x^2 - 1) = \sqrt{(x^2 - 1) + 1} = \sqrt{x^2} = |x|
   \]
   
   So, \( f \circ g(x) = |x| \).

2. **Domain \( D(f \circ g) \)**: Since \( f \circ g(x) = |x| \) is defined for all real numbers, the domain \( D(f \circ g) \) is:
   \[
   D(f \circ g) = (-\infty, \infty)
   \]

3. **\( g \circ f \)**: The composition \( g \circ f(x) = g(f(x)) \).
   \[
   g(f(x)) = g(\sqrt{x + 1}) = (\sqrt{x + 1})^2 - 1 = x + 1 - 1 = x
   \]
   
   So, \( g \circ f(x) = x \).

4. **Domain \( D(g \circ f) \)**: For \( g \circ f(x) = x \), we need \( f(x) = \sqrt{x + 1} \) to be defined, which requires \( x + 1 \geq 0 \Rightarrow x \geq -1 \). Thus, the domain \( D(g \circ f) \) is:
   \[
   D(g \circ f) = [-1, \infty)
   \]

### Problem 3
Given:
\[
y = e^{-x^2}
\]
Determine the domain \( D(f) \), range \( R(f) \), and sketch the graph of the function.

1. **Domain \( D(f) \)**: Since \( e^{-x^2} \) is defined for all real values of \( x \), the domain \( D(f) \) is:
   \[
   D(f) = (-\infty, \infty)
   \]

2. **Range \( R(f) \)**: Since \( e^{-x^2} \) is an exponential function with a maximum value of 1 (when \( x = 0 \)) and approaches 0 as \( x \) approaches \( \pm \infty \), the range is:
   \[
   R(f) = (0, 1]
   \]

3. **Graph**: The graph of \( y = e^{-x^2} \) is a bell-shaped curve symmetric about the y-axis, peaking at \( y = 1 \) when \( x = 0 \) and gradually approaching \( y = 0 \) as \( x \) moves away from zero.

Would you like further details on any of these solutions or help with visualizing the graph? 

### Related Questions
1. How do you determine the range of a square root function?
2. Why is the domain for composite functions sometimes more restricted?
3. What are typical methods for finding the range of exponential functions?
4. How does the square root function affect the domain of compositions?
5. What makes \( e^{-x^2} \) different from typical exponential functions?

### Tip
When determining the range of functions involving square roots or absolute values, carefully consider how these operations limit the possible output values.

1. Diketahui f(x) = 1 - √(x - 3). Tentukan D(f) dan R(f)!
2. Diketahui f(x) = √(x + 1) dan g(x) = x^2 - 1. Tentukan:
a. f ∘ g dan D(f ∘ g)
b. g ∘ f dan D(g ∘ f)
3. Diketahui fungsi y = e^(-x^2). Tentukan D(f), R(f), dan gambar grafik fungsi tersebut!

Solve questions on determining the domain and range of functions, including square root, exponential, and composite functions. Detailed solutions for f(x) = 1 - √(x - 3), compositions f ∘ g and g ∘ f, and y = e^(-x^2). Suitable for high school students in Grades 11-12.

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