Find the domain and range for each of the six given graphs.

Let's analyze each graph one by one to determine the domain and range.

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🔹 Definitions:

- Domain: The set of all possible x-values (horizontal extent) for which the function is defined.
- Range: The set of all possible y-values (vertical extent) that the function takes.

We'll use interval notation, considering whether endpoints are included (closed circle = included, open circle = not included).

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1) Graph 1: Parabola opening upward

- Shape: A parabola opening upwards with vertex at approximately $(-1, -4)$
- Left end: Open circle at $x = -3$, so $x > -3$
- Right end: Extends to infinity on the right

✔ Domain: All x-values from $-3$ to $\infty$, but not including -3
→ $ \boxed{(-3, \infty)} $

✔ Range: Minimum y-value is $-4$ at the vertex. Since it opens up, y goes to $\infty$
→ $ \boxed{[-4, \infty)} $

> Note: Closed circle at vertex → includes $y = -4$

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2) Graph 2: Cubic-like curve

- Starts at $x = -3$, open circle → not included
- Ends at $x = 2$, open circle → not included
- Curve goes from top left to bottom right, passing through origin

✔ Domain: From $-3$ to $2$, excluding both ends
→ $ \boxed{(-3, 2)} $

✔ Range: Highest point near $y = 5$, lowest near $y = -5$, and since it's continuous and covers all values in between
→ $ \boxed{(-5, 5)} $

> Both ends are open, so no inclusion of $y = -5$ or $y = 5$

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3) Graph 3: Hyperbolic shape (likely rational function)

- Two branches: one on the left ($x < 0$), one on the right ($x > 0$)
- Vertical asymptote at $x = 0$ — undefined there
- As $x \to 0^-$, $y \to \infty$; as $x \to 0^+$, $y \to -\infty$
- Left branch: $x < 0$, $y > 0$
- Right branch: $x > 0$, $y < 0$

✔ Domain: All real numbers except $x = 0$
→ $ \boxed{(-\infty, 0) \cup (0, \infty)} $

✔ Range: On left side, $y > 0$; on right side, $y < 0$. So all real numbers except $y = 0$
→ $ \boxed{(-\infty, 0) \cup (0, \infty)} $

> Note: Never touches $y = 0$

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4) Graph 4: Line segment with open endpoint

- Line starts at $x = -4$, closed circle → included
- Ends at $x = 4$, open circle → not included
- Slope is positive, passes through points like $(-4, -2)$ to $(4, 4)$

✔ Domain: From $x = -4$ to $x = 4$, not including 4
→ $ \boxed{[-4, 4)} $

✔ Range: At $x = -4$, $y = -2$ (closed); at $x \to 4^-$, $y \to 4$ (open)
So $y$ goes from $-2$ to $4$, not including 4
→ $ \boxed{[-2, 4)} $

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5) Graph 5: Downward parabola with endpoints

- Vertex at $(0, 2)$
- Left endpoint: closed circle at $(-3, -4)$
- Right endpoint: open circle at $(2, -5)$
- Opens downward

✔ Domain: From $x = -3$ to $x = 2$, including -3, excluding 2
→ $ \boxed{[-3, 2)} $

✔ Range: Maximum $y = 2$ at vertex, minimum $y = -5$ at right end (but open circle → not included). So $y$ goes from $-5$ to $2$, not including -5? Wait — check:

Wait! The left endpoint is at $(-3, -4)$, closed, so $y = -4$ is included.
The right endpoint is $(2, -5)$, open, so $y = -5$ is not included.

But the lowest point is at the right end — so the range is from $-5$ (not included) up to $2$ (included).

So:

- Minimum $y$: approaches $-5$ but never reaches it
- Maximum $y$: $2$ at vertex (included)

So Range: $ \boxed{(-5, 2]} $

✔ Domain: $ \boxed{[-3, 2)} $

✔ Range: $ \boxed{(-5, 2]} $

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6) Graph 6: V-shaped absolute value function

- Vertex at origin $(0, 0)$
- Lines go to $(-4, 5)$ and $(4, 5)$, both closed circles
- Symmetric V-shape

✔ Domain: From $x = -4$ to $x = 4$, both endpoints included
→ $ \boxed{[-4, 4]} $

✔ Range: Minimum $y = 0$ at vertex, maximum $y = 5$ at both ends
→ $ \boxed{[0, 5]} $

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✔ Final Answers:

| Graph | Domain | Range |
|------|--------|-------|
| 1) | $(-3, \infty)$ | $[-4, \infty)$ |
| 2) | $(-3, 2)$ | $(-5, 5)$ |
| 3) | $(-\infty, 0) \cup (0, \infty)$ | $(-\infty, 0) \cup (0, \infty)$ |
| 4) | $[-4, 4)$ | $[-2, 4)$ |
| 5) | $[-3, 2)$ | $(-5, 2]$ |
| 6) | $[-4, 4]$ | $[0, 5]$ |

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Let me know if you'd like these written out in words or need help interpreting any part!