Question

select the correct answer.
ronnie took a survey of eight of his classmates about the number of siblings they have and the number of pets they have. a table of his results is below.

# of siblings	# of pets
1	3
0	7
2	4
4	6
1	2
5	8
3	3

which of the following does ronnie’s data represent?
a. a relation only
b. neither a relation nor a function
c. a function only
d. both a function and a relation

Explanation:

Step1: Recall definitions of relation and function

A relation is a set of ordered pairs. A function is a relation where each input (x - value) has exactly one output (y - value). Here, let the number of siblings be the input (x) and the number of pets be the output (y).

Step2: Check if it is a relation

The data is a set of ordered pairs (siblings, pets) like (3,4), (1,3), (0,7), etc. So it is a relation.

Step3: Check if it is a function

Now, check for repeated x - values (number of siblings) and see if their y - values (number of pets) are the same.
- For x = 1: we have (1,3) and (1,2) → different y - values.
- For x = 3: we have (3,4) and (3,3) → different y - values.
Since there are x - values with multiple y - values, it does not satisfy the definition of a function. But wait, wait, no—wait, the input here: wait, maybe I mixed up. Wait, the table is (# of siblings, # of pets). So the domain is the number of siblings, codomain is number of pets. Let's list the ordered pairs:
(3,4), (1,3), (0,7), (2,4), (4,6), (1,2), (5,8), (3,3)
Now, check for each x (siblings) how many y (pets) it maps to:
- x = 1: maps to 3 and 2 → two different y - values.
- x = 3: maps to 4 and 3 → two different y - values.
So, since a function requires that each x has exactly one y, this is not a function. But wait, no—wait, the question is: is it a relation, a function, both, or neither?
Wait, a relation is any set of ordered pairs. So it is a relation. But is it a function? No, because of the repeated x's with different y's. Wait, but wait—maybe I made a mistake. Wait, let's re - examine. Wait, the definition of a relation is just a set of ordered pairs (or a mapping between two sets). A function is a special type of relation where each element in the domain has exactly one image in the codomain. So since there are elements in the domain (number of siblings) that have more than one image (number of pets), it is a relation (because it is a set of ordered pairs) but not a function. Wait, but wait the options:
A. a relation only
B. neither
C. a function only
D. both

Wait, but according to the ordered pairs, it is a relation (since it's a set of ordered pairs). But is it a function? No, because x = 1 has two y's (3 and 2), x = 3 has two y's (4 and 3). So it is a relation but not a function. So the answer should be A? Wait, no—wait, maybe I mixed up the input and output. Wait, maybe the input is the number of pets? No, the table is # of siblings (first column) and # of pets (second column). So the ordered pairs are (siblings, pets). So domain is siblings, range is pets. So as a relation, it's a set of ordered pairs. As a function, no, because of the repeated domain elements with different range elements. So it is a relation only. Wait, but let's check the options again. Option A: a relation only. Option D: both. Wait, maybe I made a mistake. Wait, let's check the definition again. A relation is any collection of ordered pairs. A function is a relation where each input has exactly one output. So if there are inputs with multiple outputs, it's a relation but not a function. So the data is a relation (since it's a set of ordered pairs) but not a function. So the answer is A? Wait, but wait the table: let's list all the (siblings, pets) pairs:

1. (3,4)
2. (1,3)
3. (0,7)
4. (2,4)
5. (4,6)
6. (1,2)
7. (5,8)
8. (3,3)

Now, for x = 1 (siblings = 1), we have y = 3 and y = 2. For x = 3 (siblings = 3), we have y = 4 and y = 3. So, since there are x - values with more than one y - value, it is not a function. But it is a relation because it is a set of ordered pairs. So the answer is A: a relation only.

Answer:

A. a relation only