Question

1. determine whether each relationship represents a function or not a function. write \function\ or

ot a function\ and explain why or why not.

a.

b.

x	y
-10	50
-5	25
0	0
5	25

c.
domain
0, 3, 6
range
-2, -1, 0

d.

Explanation:

Part a:

Step 1: Recall the vertical line test.

A graph represents a function if any vertical line drawn intersects the graph at most once.

Step 2: Apply the vertical line test to the graph.

For the given graph, when we draw a vertical line (parallel to the y - axis), it will intersect the graph at only one point. So, by the vertical line test, this relationship is a function.

Part b:

Step 1: Recall the definition of a function for a table.

A table represents a function if each input (x - value) has exactly one output (y - value).

Step 2: Check the x - values in the table.

The x - values are - 10, - 5, 0, 5. Each x - value has a unique y - value (even though 25 is repeated for different x - values, the key is that each x has one y). So, this relationship is a function.

Part c:

Step 1: Recall the definition of a function for a mapping.

A mapping represents a function if each element in the domain (input) is mapped to exactly one element in the range (output).

Step 2: Check the domain elements.

The domain element 0 is mapped to - 2 (only one mapping). The domain element 3 is mapped to - 2 (wait, no, looking at the mapping: 0 is mapped to - 2, 3 is mapped to - 1? Wait, no, the mapping: 0 has an arrow to - 2, 3 has an arrow to - 2? Wait, no, the diagram: 0 is connected to - 2, 3 is connected to - 2? Wait, no, let's re - examine. The domain has 0, 3, 6. 0 is connected to - 2, 3 is connected to - 2? Wait, no, the original mapping: 0→ - 2, 3→ - 2? Wait, no, the user's diagram: Domain {0,3,6}, Range {- 2, - 1, 0}. 0 is connected to - 2, 3 is connected to - 2? Wait, no, maybe I misread. Wait, 0 has an arrow to - 2, 3 has an arrow to - 1? Wait, no, the lines: 0 to - 2, 0 to - 1? No, the diagram: 0 is connected to - 2, 3 is connected to - 2, and 6 is connected to - 0? Wait, no, the correct way: In a function, each domain element must have exactly one range element. Here, the domain element 0 is mapped to - 2 (only one? Wait, no, if 0 is connected to two range elements, then it's not a function. Wait, looking at the diagram: 0 has two arrows? Wait, the user's diagram: 0 is connected to - 2, and 0 is connected to - 1? No, maybe the diagram is 0→ - 2, 3→ - 2, 6→ - 0? Wait, no, the correct analysis: If a domain element is mapped to more than one range element, it's not a function. Wait, in the given mapping, the domain element 0 is mapped to - 2 (only one? Wait, maybe I made a mistake. Wait, the domain is {0,3,6}. 0 is connected to - 2, 3 is connected to - 2, and 6 is connected to 0? No, the lines: 0 to - 2, 0 to - 1? No, the original problem's diagram: Let's see, the domain circle has 0, 3, 6. The range circle has - 2, - 1, 0. The arrows: 0→ - 2, 0→ - 1? No, maybe the 0 is connected to - 2, 3 is connected to - 2, and 6 is connected to 0? Wait, no, the key is: For a function, each input (domain) has exactly one output (range). If 0 is mapped to two different range elements, then it's not a function. Wait, looking at the diagram, 0 has two arrows? If so, then the domain element 0 is associated with more than one range element, so this relationship is not a function.

Part d:

Step 1: Recall the vertical line test.

A graph represents a function if any vertical line drawn intersects the graph at most once.

Step 2: Apply the vertical line test to the graph.

In the given graph, if we draw a vertical line (for example, the y - axis), it will intersect the graph at more than one point. So, by the vertical line test, this relationship is not a function.

Answer:

s:
a. Function, because the vertical line test is satisfied (each vertical line intersects the graph at most once).
b. Function, because each x - value has exactly one y - value.
c. Not a function, because the domain element 0 is mapped to more than one range element (violates the definition of a function).
d. Not a function, because the vertical line test fails (a vertical line intersects the graph at more than one point).