Sure, let's address each part of the problem step-by-step.
### Step 8:
Use the information in the table to write 5 ordered pairs that have cats as the input value and dogs as the output value.
From the table:
- When there are 0 cats, there are 15 dogs.
- When there are 4 cats, there are 14 dogs.
- When there are 8 cats, there are 13 dogs.
- When there are 12 cats, there are 12 dogs.
- When there are 16 cats, there are 11 dogs.
These give us the ordered pairs:
[tex]\[
(0, 15), (4, 14), (8, 13), (12, 12), (16, 11)
\][/tex]
### Step 9:
Write an explicit equation that shows how many dogs they can accommodate based on how many cats they have. The number of dogs [tex]\((d)\)[/tex] will be a function of the number of cats [tex]\((c)\)[/tex], or [tex]\(d=f(c)\)[/tex].
From the table:
- The number of dogs decreases by 1 for every 4 cats added.
- At 0 cats, there are 15 dogs. With an increase of 4 cats, the count of dogs decreases by 1.
So, the relationship appears linear and can be represented as:
[tex]\[
d = 15 - \frac{c}{4}
\][/tex]
### Step 10:
Use the information in the table to write 5 ordered pairs that have dogs as the input value and cats as the output value.
From the table:
- When there are 15 dogs, there are 0 cats.
- When there are 14 dogs, there are 4 cats.
- When there are 13 dogs, there are 8 cats.
- When there are 12 dogs, there are 12 cats.
- When there are 11 dogs, there are 16 cats.
These give us the ordered pairs:
[tex]\[
(15, 0), (14, 4), (13, 8), (12, 12), (11, 16)
\][/tex]
### Step 11:
Write an explicit equation that shows how many cats they can accommodate based on how many dogs they have. The number of cats [tex]\((c)\)[/tex] will be a function of the number of dogs [tex]\((d)\)[/tex], or [tex]\(c=g(d)\)[/tex].
To express the relationship from the equation in Step 9:
If [tex]\( d = 15 - \frac{c}{4} \)[/tex], we solve for [tex]\( c \)[/tex]:
[tex]\[
c = 4(15 - d)
\][/tex]
### Step 12:
Describe how the ordered pairs from Step 8 and Step 10 are different.
In Step 8, the pairs have cats as the input and dogs as the output. In Step 10, the pairs have dogs as the input and cats as the output. Essentially, the pairs from Step 8 swap the positions of their elements compared to the pairs from Step 10, i.e., [tex]\( (cat, dog) \)[/tex] becomes [tex]\( (dog, cat) \)[/tex].
### Step 13:
a. Describe the domain for [tex]\( d=f(c) \)[/tex].
For the function [tex]\( d = 15 - \frac{c}{4} \)[/tex]:
- The table provides cat values from 0 to 60.
- Therefore, the domain for [tex]\( c \)[/tex] in [tex]\( f(c) \)[/tex] is:
[tex]\[
0 \le c \le 60
\][/tex]
- Since [tex]\( d \)[/tex] has to remain non-negative, the maximum value [tex]\( c \)[/tex] can reach, calculated by setting [tex]\( d = 0 \)[/tex], is:
[tex]\[
c = 60
\][/tex]
So, the domain of [tex]\( d = 15 - \frac{c}{4} \)[/tex] is [tex]\( 0 \le c \le 60 \)[/tex].
