Let's find the function B(p) first. Then we can compare the domains, ranges, slopes, and y-intercepts of the functions. Finally, we will interpret the comparisons in context.
Finding B(p)
From the table, we can see that B(0)=24. Therefore, the y-intercept of B(p) is 24. So far, we have the following function.
B(p)=mp+24
To find m, we can substitute one of the given points in the equation. Let's take (50,30).
B(p)=mp+24
30=m ( 50)+24
6=m ( 50)
6/50=m
m=6/50
m=0.12
The slope is 0.12. We can write the final equation as:
B(p)=0.12p+24.
Domain
Domain is the set of all possible inputs p. We are given these sets for each function in the exercise. For both A(p) and B(p), this is the set of whole numbers p, where
0 < p ≤ 500.
Range
Since A(p) and B(p) are linear functions, both with the domain 0 < p ≤ 500, the ranges of A(p) and B(p) will be:
A(0)< A(p)≤ A(500)
and
B(0)< B(p)≤ B(500).
We know that A(0) and B(0) represent the y-intercepts of the functions. Therefore,
A(0)=15
and
B(0)=24.
Let's find A(500) and B(500) by substituting p=500 into each function, beginning with A(p)!
A(p)=0.13p+15
A( 500)=0.13 ( 500) +15
A(500)=65+15
A(500)=80
Now let's check B(p).
B(p)=0.12p+24
B( 500)=0.12 ( 500) +24
B(500)=60+24
B(500)=84
We can conclude that the ranges of A(p) and B(p) are:
15 < A(p) ≤ 80
24 < B(p) ≤ 84
Slope and y-intercept
Next, let's compare the slopes and y-intercepts!
A(p)&= 0.13p+ 15
B(p)&= 0.12p+ 24
The slope of A(p) is greater than the slope of B(p). However, the y-intercept of A(p) is less than the y-intercept of B(p).
Interpretation
Domains
The domain 0 < p ≤ 500, tells us that a textbook for both colleges can have the maximum of 500 pages.
Ranges
The range of each function is
&15 < A(p) ≤ 80
&24 < B(p) ≤ 84.
Therefore, the cost of producing textbooks for college A is at least $15 and maximum $80. The range of B(p) means that the cost of producing textbooks for college B is at least $24 and maximum $84.
Slopes
The slopes are:
0.13 and 0.12.
The cost increases by the value of slope as the number of pages increases. Therefore, the slopes represent the cost of printing one page for each college. The cost per page for college A is greater than the cost for college B.
y-intercepts
Lastly, the y-intercepts are:
15 and24.
This is an additional cost that each college charges for the production of each textbook. This is probably some kind of one-time service fee. It is $15 for college A and $24 for college B. Thus, the flat rate cost at college B is greater than the cost at college A.
