Name: ________________________
Composition of Functions Learning Task
Part I: Is Your Heart Rate Normal for that Medication?
Many medications have side effects that doctors must consider when prescribing the medication. For
example, many cold medications have the effect of raising the takers blood pressure. Therefore, even when
recommending over the counter cold medications, doctors and pharmacists must consider whether the
patient is already on medication for high blood pressure. If the patient is on such a medication, then the cold
medicine recommend will have to be one that does not affect blood pressure.
In particular, some medications can increase the heart rate, or beats per minute, of a patient of normal health
in a predictable manner depending on how much medication is in the patient’s system. Based on research
and study of one medication, we shall call only D, one such set of data is given in the table 1:
D, drug level
in mg
r, heart rate in
beats per minute
0
50
100
150
200
250
60
70
80
90
100
110
Write a function r(D) that describes the relationship between the amount of drug administered and the
resulting heart rate.
What does this function tell us about the relationship between the amount of drug administered and the
resulting heart rate? What general conclusions can we draw?
The data in table 1 represents what happens with the initial introduction of the medication. As time passes,
the medication is processed by the body and the effect will lessen. Table 2 gives the amount of drug D in
the patient’s body after t, hours from injection:
t, time in hours
since injection
0
1
2
3
4
5
6
7
8
D, drug level in
mg
250
200
160
128
102
82
66
52
42
Write a function D(t) that describes the relationship between the amount of drug in the patient’s system and
the elapsed time since injection.
What does this function tell us about the relationship between the amount of drug in the patient’s system and
the elapsed time since injection? What general conclusions can we draw?
GPS Pre-Calculus
1st Edition
Unit 8
However, to monitor a patient to be sure that the effect of the drug is as expected, the medical
professionals take the patient’s pulse or heart rate. Therefore, we need to find a table and a function
that will predict the patient’s heart rate as time since injection. Using the data given in Tables 1 and 2,
complete the table below:
t, time in hours
0
1
2
3
4
5
6
7
8
r, heart rate in
beats per minute
Now, find a function that would model this data? How could such a function be formed? In other
words, we need r(t). How do we find it?
What kind of function is r(t)? How do you describe its graph? How does it’s graph relate to the
graphs of the r(D) and D(t) functions? Graph all three on the same window. What relationship exists?
Use r(t) to find the predicted heart rate of a patient who was given the medication D 3 hours ago. How
would you use this information to determine if the patient is having an abnormal reaction to the drug?
Georgia Department of Education, State Superintendent of Schools
Copyright 2011© All Rights Reserved
Unit 8: Page 2 of 8
GPS Pre-Calculus
1st Edition
Unit 8
Part II Interpreting a Table
As the cost of a post-secondary education rises, many students are working jobs to help pay for
college. However, these jobs often affect the number of hours they have to study, so may affect the
number of credit hours they pursue during a semester. Table 1 gives such an example of the effect of
the numbers worked on the number of credits attempted. Table 2 gives an example of the number of
hours needed to study relative to the number of hours attempted.
Table 1
Table 2
Number of
hours worked
per week
Number of
credits
attempted
Number of
credits
attempted
Number of hours
needed to study
per week
0  h 4
18
12
12
4  h 8
17
13
14
8  h  12
16
14
16
12  h  16
15
15
20
16  h  20
14
16
25
20  h  24
13
17
31
24  h  28
12
18
39
Graph Table 1 by hand on grid paper. What type of function does it appear to be? What transformations
need to be made to the basic function to model the data given? Write a function rule for this table. Call this
function C(w).
Graph Table 2 on a graphing utility or on grid paper. What type of function does it appear to be? What
transformations need to be made to the basic function to model the data given? Write a function rule for this
table. Call this function S(C).
Georgia Department of Education, State Superintendent of Schools
Copyright 2011© All Rights Reserved
Unit 8: Page 3 of 3
GPS Pre-Calculus
Unit 8
1st Edition
Complete the table below:
Number of
hours worked
per week
Number of hours
needed to study
per week
0  h 4
4  h 8
8  h  12
12  h  16
16  h  20
20  h  24
24  h  28
As the number of hours worked increases, what happens to the number of hours spent studying?
Graph this data on grid paper. What type of function does it appear to be? How does this function
relate to the ones from Tables 1 and 2? Would the use of the functions you wrote for Tables 1 and 2
help to find a function rule for this data? Write the function that relates number of hours spent
studying to the number of hours worked per week. Call this function S(w).
Georgia Department of Education, State Superintendent of Schools
Copyright 2011© All Rights Reserved
Unit 8: Page 4 of 8
GPS Pre-Calculus
1st Edition
Unit 8
Notes on Composition of Functions Learning Task
Composition of Functions Learning Task
Part I Is Your Heart Rate Normal for that Medication?
Many medications have side effects that doctors must consider when prescribing the medication. For
example, many cold medications have the effect of raising the takers blood pressure. Therefore, even when
recommending over the counter cold medications, doctors and pharmacist must consider whether the patient
is already on medication for high blood pressure. If the patient is on such a medication, then the cold
medicine recommend will have to be one that does not affect blood pressure.
In particular, some medications can increase the heart rate, or beats per minute, of a patient of normal health
in a predictable manner depending on how much medication is in the patient’s system. Based on research
and study of one medication, we shall call only D, one such set of data is given in the table 1.
D, drug level
in mg
0
50
100
150
200
250
r, heart rate in beats
per minute
60
70
80
90
100
110
Write a function r(D) that describes the relationship between the amount of drug administered and the
resulting heart rate.
r(D) = .2 D + 60
What does this function tell us about the relationship between the amount of drug administered and the
resulting heart rate? What general conclusions can we draw?
This is a linear function so the increase in the heart rate increases at a constant relative to the amount of
drug given. As D increase so does r at a constant rate.
The data in table 1 represents what happens with the initial introduction of the medication. As time passes,
the medication is processed by the body and the effect will lessen. Table 2 gives the amount of drug D in
the patient’s body after t, hours from injection.
t, time in hours
since injection
0
1
2
3
4
5
6
7
8
D, drug level in
mg
250
200
160
128
102
82
66
52
42
Write a function D(t) that describes the relationship between the amount of drug in the patient’s system and
the elapsed time since injection.
D(t) = 250(.8)t
Georgia Department of Education, State Superintendent of Schools
Copyright 2011© All Rights Reserved
Unit 8: Page 5 of 3
GPS Pre-Calculus
1st Edition
Unit 8
What does this function tell us about the relationship between the amount of drug in the patient’s system and
the elapsed time since injection? What general conclusions can we draw?
This is an exponential function so the decrease in the amount of drug is geometrically relative to the time
since injection. As t increases, D decreases at a variable rate.
However, to monitor a patient to be sure that the effect of the drug is as expected, the medical professionals
take the patient’s pulse or heart rate. Therefore, we need to find a table and a function that will predict the
patient’s heart rate as time since injection. Using the data given in Tables 1 and 2, complete the table
below:
t, time in hours
0
1
2
3
4
5
6
7
8
r, heart rate in beats
per minute
Now, find a function that would model this data? How could such a function be formed? In other words,
we need r(t). How do we find it?
Students have already been exposed to the idea of composition of functions in GPS Algebra as it relates
to verifying inverse functions. They may or may not remember this term, but they should be able to
express the idea of substituting one function into the other. Hopefully, the notation and units that have
been used to describe the problem situation will lead them to this conclusion.
Since the first table gives rate dependent on Drug, r(D), and the second table gives Drug dependent on
time, D(t), then r(t) = r(D(t)), the composition of the two functions.
So, r(t) = .2( 250(.8)t) + 60.
What kind of function is r(t)? How do you describe its graph? How does its graph relate to the graphs of
the r(D) and D(t) functions? Graph all three on the same window. What relationship exists?
Use r(t) to find the predicted heart rate of a patient who was given the medication D 3 hours ago. How
would you use this information to determine if the patient is having an abnormal reaction to the drug?
Part II Interpreting a Table
As the cost of a post-secondary education rises, many students are working jobs to help pay for college.
However, these jobs often affect the number of hours they have to study, so may affect the number of credit
hours they pursue during a semester. Table 1 gives such an example of the effect of the numbers worked on
the number of credits attempted. Table 2 gives an example of the number of hours needed to study relative
to the number of hours attempted.
Table 1
Table 2
Number of
hours worked
per week
Number of
credits
attempted
Number of
credits
attempted
Number of hours
needed to study
per week
0  h 4
18
12
12
Georgia Department of Education, State Superintendent of Schools
Copyright 2011© All Rights Reserved
Unit 8: Page 6 of 8
GPS Pre-Calculus
1st Edition
Unit 8
Number of
hours worked
per week
Number of
credits
attempted
Number of
credits
attempted
Number of hours
needed to study
per week
4  h 8
17
13
14
8  h  12
16
14
16
12  h  16
15
15
20
16  h  20
14
16
25
20  h  24
13
17
31
24  h  28
12
18
39
Graph Table 1 by hand on grid paper. What type of function does it appear to be? What transformations
need to be made to the basic function to model the data given? Write a function rule for this table. Call this
function C(w).
The graph of the data should look like a transformation of the greatest integer function.
However, the segments are stretched by a factor of four, they have been shifted vertically by 18 and the
steps decrease by one instead of increase by one. Therefore, as w increases, C decreases.
w
C(w) = 18   
4
Graph Table 2 on a graphing utility or on grid paper. What type of function does it appear to be? What
transformations need to be made to the basic function to model the data given? Write a function rule for this
table. Call this function S(C).
The graph of the data should look like a transformation of an exponential function. It has been stretched
vertically by a factor of 1.046 and horizontally by a factor of 1. 22. As C increases, S increases.
S(C) = 1.046(1.221)C
Complete the table below:
Number of
hours worked
per week
Number of hours
needed to study
per week
0  h 4
4  h 8
8  h  12
Georgia Department of Education, State Superintendent of Schools
Copyright 2011© All Rights Reserved
Unit 8: Page 7 of 3
GPS Pre-Calculus
Number of
hours worked
per week
1st Edition
Unit 8
Number of hours
needed to study
per week
12  h  16
16  h  20
20  h  24
24  h  28
As the number of hours worked increases, what happens to the number of hours spent studying?
Graph this data on grid paper. What type of function does it appear to be? How does this function relate to
the ones from Tables 1 and 2? Would the use of the functions you wrote for Tables 1 and 2 help to find a
function rule for this data? Write the function that relates number of hours spent studying to the number of
hours worked per week. Call this function S(w).
This graph is actually a step function where the location of each step is determined by an exponential
decay function.
w
(18    )
w
 
w
 
S(w) = S(C(w)) = 1.046(1.221)  4  = 1.046(1.221)18 (1.2211 ) 4  = 38.055(.819) 4 
Students at this point should be able to step through the transformations, the algebra, and the regression
used in this solution. However, students will need to see that the relationship between the tables means
the entries are reversed when moving from Table 2 into the new table. This is why the base of the
exponential changed from above to below 1. These characteristics that result from the composition and
transformation are important to have students discuss.
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Copyright 2011© All Rights Reserved
Unit 8: Page 8 of 8