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?

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.