Let's go step by step through each part of your problem.

1. Domain and Range of Each Function

There isn't a specific function listed in "1. 2." in your query. If you'd like to provide the equations or graphs, I can help determine the domain and range for each.

3. Is the Relation a Function? Is it One-to-One or Many-to-One?

Relation: {(–4, 7), (–3, 5), (1, 4), (3, –8), (5, –11)}

• A function requires that each input (x-value) corresponds to exactly one output (y-value). In this case, all the x-values are different, meaning there are no duplicate inputs.
• Therefore, this relation is a function.
• One-to-One or Many-to-One: In a one-to-one function, each output (y-value) is associated with only one input (x-value). Since no y-value is repeated in the relation, this is a one-to-one function.

4. Is the Relation a Function? Is it One-to-One or Many-to-One?

Relation: {(–4, 8), (–2, 4), (0, 1), (2, 4), (4, 8)}

• This is a function because each x-value corresponds to exactly one y-value (no duplicate inputs).
• However, the y-values are repeated: (–4, 8) and (4, 8) have the same output, as do (–2, 4) and (2, 4).
• Therefore, this is a function, but it is a many-to-one function because multiple x-values share the same y-value.

5. Relation (table format) Interpretation

It seems like you're presenting a table of inputs and outputs:

x	y
-2	-3
0	1
1	3
3	4

• This is a function since each x-value has a unique y-value.
• All the outputs (y-values) are distinct, so this is a one-to-one function.

6. Graph Interpretation

You're referencing a graph but without the visual, it's difficult to interpret. From the table, the function appears to map x-values like -4, 0, and 4 to corresponding y-values like 4, 8, etc. If you'd like, you can upload a photo of the graph and I can provide detailed insights.

7. Vertical Line Test Explanation

The vertical line test is used to determine whether a relation is a function. If a vertical line intersects a graph more than once, the relation is not a function because this would indicate that a single input (x-value) has multiple outputs (y-values).

• If a vertical line touches a graph at exactly one point wherever you draw it, the relation is a function.
• If the line touches the graph more than once, the relation fails the test and is not a function.

8. Fiona's Gas Graph and Domain Constraints

Fiona buys gas at $2.25 per unit. If we assume the graph shows the amount of gas purchased (x-axis) versus the cost (y-axis):

• The domain represents the amount of gas purchased. The constraints are:
  • She cannot purchase a negative amount of gas, so the domain must be non-negative (x ≥ 0).
  • There may be a maximum limit on the amount of gas she can buy (based on the car's tank size), but that's not specified here.

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Would you like to explore more details on any of these points? Here are some related questions to deepen your understanding:

1. How do you find the domain and range of a given function graphically?
2. What is the difference between one-to-one and many-to-one functions, and why does it matter?
3. How can we apply the vertical line test to real-world problems?
4. What are the practical implications of a function being many-to-one?
5. How can constraints in real-world scenarios (like gas purchases) affect the domain of a function?

Tip: The domain of a function is typically the set of all possible inputs, while the range is the set of all possible outputs. Always pay attention to restrictions on these sets!