https://file.aiquickdraw.com/mathbot/file/17284531234506qradms7.jpg

Let's analyze the problems from the image.

### Problem 5:
The graph appears to represent a cubic function, likely in the form of \( y = x^3 \). The function continues indefinitely in both directions on the x-axis and y-axis.

- **Domain**: All real numbers \( (-\infty, \infty) \)
- **Range**: All real numbers \( (-\infty, \infty) \)

### Problem 6:
This is a parabolic graph, likely representing a quadratic function such as \( y = -x^2 + 2 \), as it is opening downwards and has a maximum at \( y = 2 \).

- **Domain**: All real numbers \( (-\infty, \infty) \)
- **Range**: \( (-\infty, 2] \), because the highest point on the y-axis is 2 and the graph opens downwards.

### Problem 7:
This graph represents a constant function where all points are horizontally aligned at \( y = 5 \).

- **Domain**: \( [2, 8] \), because the x-values range from 2 to 8.
- **Range**: \( \{5\} \), a single value because the y-value is constant at 5.

### Problem 8:
This is an upside-down parabola, similar to a quadratic function with a maximum. The vertex appears to be at \( (0, -2) \), and the parabola opens downward.

- **Domain**: \( [-4, 4] \) since the graph extends from \( x = -4 \) to \( x = 4 \).
- **Range**: \( [-6, -2] \), as the minimum value on the y-axis is -6, and the maximum value is -2.

### Problem 9:
- **Expression**: "The sum of the square of \( x \) and the square of the opposite of \( x \)":
  
  This is the same as calculating \( x^2 + (-x)^2 \). Since both \( x^2 \) and \( (-x)^2 \) are equal to \( x^2 \), the expression simplifies to:
  \[
  x^2 + x^2 = 2x^2
  \]
  Therefore, it is not equivalent to 0 but is equivalent to \( 2x^2 \).

Would you like any further clarification or additional details?

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Here are 5 related questions:
1. What would the domain and range be for a cubic function like \( y = x^3 - 4 \)?
2. How would the domain change if the parabola in problem 6 were translated to the left or right?
3. How can you determine the range of a parabola from its vertex form equation?
4. Why do quadratic functions sometimes have a limited range while cubic functions do not?
5. What happens to the graph of a constant function if we change its y-value?

**Tip**: The range of a function is closely tied to its highest and lowest points (like the vertex of a parabola), while the domain is usually determined by how far the graph extends along the x-axis.

Find the domain and range of the function represented by the graph, and determine if the expression 'the sum of the square of x and the square of the opposite of x' is equivalent to 0 or 2x².

Learn how to determine the domain and range of various functions using graphs and analyze the expression involving the sum of squares. This problem covers key algebraic concepts, including quadratic, cubic, and constant functions, and graph analysis. Suitable for students in Grades 9-11.

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