Determine if each relation is a function or not a function.

To determine if a relation is a function, we use the Vertical Line Test for graphs and check for unique inputs in tables, sets, and mapping diagrams.

Rule: A relation is a function if every input ($x$) has exactly one output ($y$). If any single $x$ value is paired with more than one $y$ value, it is not a function.

Here is the step-by-step sorting of the items:

1. Tables

* Top Left Table:
* $x$: 3, 3, 4, 4, 5
* The input $3$ goes to both $-1$ and $0$. The input $4$ goes to both $-1$ and $2$.
* Result: NOT A FUNCTION
* Top Right Table:
* $x$: -1, -2, -3, -4, -5
* All $x$ values are different. Each has only one $y$.
* Result: FUNCTION
* Middle Right Table:
* $x$: -1, -1, -1, -1, -1
* The input $-1$ goes to many different outputs ($3, 4, -2, 2, 0$).
* Result: NOT A FUNCTION
* Bottom Left Table:
* $x$: 1, 4, 6, 8, 1
* The input $1$ appears twice with different outputs ($-2$ and $-2$... wait, let's look closer. Actually, usually duplicates with same output are okay, but here the last pair is $(1, -2)$ and first is $(1, -2)$. However, looking at the pattern of these problems, usually distinct x's are required or unique mappings. Let's re-read carefully. Input 1 maps to -2. Input 1 maps to -2. This is technically consistent. BUT, looking at the other "Not a Function" examples, they have clear conflicts. Let's look at the bottom middle table first.
* Correction/Refinement: Let's look at the Bottom Middle Table:
* $x$: -2, -3, -4, -5. All unique.
* Result: FUNCTION
* Now back to Bottom Left Table: Inputs are 1, 4, 6, 8, 1. Outputs are -2, 0, -2, 0, -2. The input 1 maps to -2 in both cases. The input 4 maps to 0. The input 6 maps to -2. The input 8 maps to 0. Since no single $x$ maps to *different* $y$'s, this is technically a function. However, in many school contexts, if an $x$ is repeated in the list, students are often tricked. Let's look at the set notation below it: $\{(-2, 1), (1, -2), (-1, 2), (-2, 1), (-3, -2)\}$. Wait, the table and the set below it might be separate cards.
* Let's treat the Table and the Set below it as separate items.
* Bottom Left Table: Inputs 1, 4, 6, 8, 1. Outputs -2, 0, -2, 0, -2. Input 1 always gives -2. Input 4 gives 0. Input 6 gives -2. Input 8 gives 0. No conflict. Result: FUNCTION. *(Self-Correction: Often in these drag-and-drops, repeated x-values in a table are a visual cue for "Not a Function" even if the y's match, OR it's a trick. Let's look at the set below it.)*
* Set below Bottom Left Table: $\{(-2, 1), (1, -2), (-1, 2), (-2, 1), (-3, -2)\}$. Here, the input $-2$ appears twice, mapping to $1$ both times. No conflict. Result: FUNCTION.
* *Alternative Interpretation:* Is it possible the table implies distinct entities? No, standard math rules say if $f(1)=-2$ and $f(1)=-2$, it's fine. But let's look at the Right Side Middle Table (black header):
* $x$: 4, 4, 5, 5.
* Input $4$ goes to $3$ AND $2$. Conflict!
* Input $5$ goes to $3$ AND $2$. Conflict!
* Result: NOT A FUNCTION.

2. Sets of Ordered Pairs

* Left Set: $\{(5, 1), (-1, 5), (-5, 1), (1, -5), (5, 3)\}$
* Input $5$ goes to $1$ AND $3$. Conflict.
* Result: NOT A FUNCTION
* Left Middle Set: $\{(1, 4), (3, 2), (-3, -2), (1, 2), (-2, -3)\}$
* Input $1$ goes to $4$ AND $2$. Conflict.
* Result: NOT A FUNCTION
* Right Middle Set: $\{(-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 0)\}$
* All $x$ values are unique. (Multiple $x$'s going to the same $y$ is allowed).
* Result: FUNCTION
* Bottom Left Set: $\{(-2, 1), (1, -2), (-1, 2), (-2, 1), (-3, -2)\}$
* Input $-2$ maps to $1$ twice. No other repeats. No conflicts.
* Result: FUNCTION

3. Graphs

* Top Middle Graph (S-shape):
* If you draw a vertical line through the middle, it hits the curve in 3 places.
* Result: NOT A FUNCTION
* Top Right Graph (Circle):
* A vertical line through the center hits the circle at the top and bottom.
* Result: NOT A FUNCTION
* Bottom Left Graph (Line):
* Any vertical line hits the straight line only once.
* Result: FUNCTION
* Bottom Right Graph (Parabola/U-shape):
* Any vertical line hits the U-shape only once.
* Result: FUNCTION

4. Mapping Diagrams (Inputs/Outputs)

* Left Mapping Diagram:
* Input $-2$ goes to $-1$ AND $3$.
* Input $4$ goes to $5$ AND $8$.
* One input has multiple outputs.
* Result: NOT A FUNCTION
* Right Mapping Diagram:
* Input $-3$ goes to $2$.
* Input $-1$ goes to $7$.
* Input $5$ goes to $7$.
* Input $8$ goes to $7$.
* Every input has exactly one arrow coming out of it. (It is okay for multiple inputs to go to the same output).
* Result: FUNCTION

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Final Sorting

FUNCTION
1. Table (Top Right): $x$ values -1 to -5 are all unique.
2. Graph (Bottom Left): The diagonal line passes the vertical line test.
3. Mapping Diagram (Right): Each input has exactly one output arrow.
4. Table (Bottom Middle): $x$ values -2, -3, -4, -5 are all unique.
5. Set (Right Middle): $\{(-3, 0)...(1, 0)\}$ All $x$'s are unique.
6. Graph (Bottom Right): The parabola passes the vertical line test.
7. Table (Bottom Left): Inputs 1, 4, 6, 8, 1. Although 1 is repeated, it maps to -2 both times. No conflict. *(Note: If your specific class marks repeated x-values in tables as "not a function" regardless of y-value, move this to Not a Function. However, mathematically, it is a function. Given the other clear errors, this is likely a Function).*
8. Set (Bottom Left): $\{(-2, 1)... \}$ Input -2 is repeated but maps to 1 both times. No conflict. Mathematically a Function.

NOT A FUNCTION
1. Table (Top Left): Input 3 maps to -1 and 0.
2. Graph (Top Middle): The S-curve fails the vertical line test.
3. Graph (Top Right): The circle fails the vertical line test.
4. Table (Top Right/Middle Right): Input -1 maps to 3, 4, -2, 2, 0.
5. Set (Top Left): Input 5 maps to 1 and 3.
6. Set (Left Middle): Input 1 maps to 4 and 2.
7. Mapping Diagram (Left): Input -2 maps to -1 and 3.
8. Table (Right Middle/Black Header): Input 4 maps to 3 and 2.

*(Self-Correction on the ambiguous items: In standardized online homework platforms like the one shown, "Tables" with repeated x-values are often tricky. However, looking at the "Not a Function" table on the right with -1 repeating 5 times, and the Top Left table with 3 and 4 repeating, the pattern is clearly about conflicting outputs. The Bottom Left table and Bottom Left Set do NOT have conflicting outputs. Therefore, they are Functions.)*

Final Answer:

FUNCTION
* Table: Top Right ($x$: -1, -2, -3, -4, -5)
* Graph: Bottom Left (Diagonal Line)
* Mapping Diagram: Right (Inputs -3, -1, 5, 8)
* Table: Bottom Middle ($x$: -2, -3, -4, -5)
* Set: Right Middle $\{(-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 0)\}$
* Graph: Bottom Right (U-shaped Parabola)
* Table: Bottom Left ($x$: 1, 4, 6, 8, 1) — *Note: Repeated x has same y.*
* Set: Bottom Left $\{(-2, 1), (1, -2), (-1, 2), (-2, 1), (-3, -2)\}$ — *Note: Repeated x has same y.*

NOT A FUNCTION
* Table: Top Left ($x$: 3, 3, 4, 4, 5) — *Conflicting outputs.*
* Graph: Top Middle (S-shaped curve)
* Graph: Top Right (Circle)
* Table: Middle Right ($x$: -1, -1, -1, -1, -1) — *Conflicting outputs.*
* Set: Top Left $\{(5, 1), (-1, 5), (-5, 1), (1, -5), (5, 3)\}$ — *Input 5 conflicts.*
* Set: Left Middle $\{(1, 4), (3, 2), (-3, -2), (1, 2), (-2, -3)\}$ — *Input 1 conflicts.*
* Mapping Diagram: Left (Inputs -2, -1, 4) — *Inputs have multiple arrows.*
* Table: Right Middle Black Header ($x$: 4, 4, 5, 5) — *Conflicting outputs.*