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The image contains four problems related to determining the domain and range of functions, including two graph-based and two equation-based questions. I'll guide you through solving each one:

### 1. Domain and Range of the Graphs

#### Graph 1:
- **Domain (D)**: The set of all x-values for which the function is defined.
  - Based on the graph, the x-values seem to span from approximately \( x = -2 \) to \( x = 2 \) (closed interval).
  - **Domain**: \( [-2, 2] \)

- **Range (R)**: The set of all y-values that the function takes.
  - The graph's y-values range from \( y = 0 \) to \( y = 3 \) (closed interval).
  - **Range**: \( [0, 3] \)

#### Graph 2:
- **Domain (D)**: The x-values span from approximately \( x = -4 \) to \( x = 4 \).
  - **Domain**: \( [-4, 4] \)

- **Range (R)**: The y-values range from \( y = -3 \) to \( y = 2 \).
  - **Range**: \( [-3, 2] \)

### 2. Equations

#### Equation 1: \( y = -3x^2 + 1 \)
- **Domain**: Since this is a quadratic function, \( x \) can take any real number.
  - **Domain**: \( (-\infty, \infty) \)
  
- **Range**: As \( x^2 \) always gives non-negative values, the function has a maximum value at \( x = 0 \), where \( y = 1 \). As \( x \) increases in either direction, \( y \) becomes negative due to the \( -3x^2 \) term.
  - **Range**: \( (-\infty, 1] \)

#### Equation 2: \( y = \frac{1}{x+3} \)
- **Domain**: The function is undefined when \( x + 3 = 0 \), or \( x = -3 \), so all real numbers except \( x = -3 \).
  - **Domain**: \( (-\infty, -3) \cup (-3, \infty) \)
  
- **Range**: Since \( y = \frac{1}{x+3} \), the function takes all real values except 0 (as the numerator is constant 1).
  - **Range**: \( (-\infty, 0) \cup (0, \infty) \)

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Would you like further details on any of these, or should I clarify a particular part? Here are five related questions:

1. What is the graphical representation of functions involving square terms like \( -3x^2 \)?
2. How do we calculate the domain for rational functions like \( y = \frac{1}{x+3} \)?
3. Why does the range of a quadratic function have a maximum or minimum value?
4. How can vertical asymptotes help determine the range of rational functions?
5. What changes to a function can affect its domain and range?

**Tip**: Always look for values that might make the denominator zero or cause square roots of negative numbers when determining domain!

The user submitted an image containing four problems related to finding the domain and range of given functions, including graph-based and equation-based questions.

This tutorial guides you through finding the domain and range for various functions, including quadratic and rational functions, using both graphical and equation-based approaches. Perfect for high school students in Grades 9-12 looking to solidify their understanding of these concepts.

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