function composition
Module 3 : Investigation 4a
MAT 170 | Precalculus
September 12, 2016
question 1
The number of calories burned while running depends on many
factors, but averages about 100 calories per mile. Suppose Nikki goes
for a run, traveling at a constant speed of 720 feet per minute and
burning 100 calories per mile that she runs.
(a) What quantities are varying in this situation ? What quantities are
constant ? (Be sure to include units.)
(c) (i) As the time (in minutes) spent running increases, how does the
distance (in feet) Nikki has traveled change ?
(c) (ii) As the distance (in feet) Nikki has traveled increases, how
does the number of calories she has burned change ?
(c) (iii) As the time (in minutes) spent running increases, how does
the number of calories she has burned change ?
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question 1 - solutions
(a) What quantities are varying in this situation ? What quantities are
constant ? (Be sure to include units.)
Varying : Amount of time Nikki has been running (in minutes),
Distance Nikki has travelled (in miles), Number of calories that
Nikki has burned.
Fixed : Nikki’s speed, Number of calories that Nikki burns per
mile.
(c) (i) As the time (in minutes) spent running increases, how does the
distance (in feet) Nikki has traveled change ?
As the time (in minutes) Nikki spends running increases, the distance (in feet) Nikki has traveled will also increase.
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question 1 - solutions
(c) (ii) As the distance (in feet) Nikki has traveled increases, how
does the number of calories she has burned change ?
As the distance (in feet) that Nikki has traveled increases, the
number of calories that Nikki has burned will also increase.
(c) (iii) As the time (in minutes) spent running increases, how does
the number of calories she has burned change ?
As the time (in minutes) Nikki spends running increases, the number of calories that Nikki has burned will also increase.
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question 1
The number of calories burned while running depends on many
factors, but averages about 100 calories per mile. Suppose Nikki goes
for a run, traveling at a constant speed of 720 feet per minute and
burning 100 calories per mile that she runs.
(d) Complete the following table :
Time (in minutes)
Nikki has been
running
Distance (in feet)
Nikki has travelled
3
720 · 3 = 2160
14
720 · 14 = 10080
18.5
720·(18.5) = 13320
22.2
720·(22.2) = 15984
t
720t
Distance (in miles)
Nikki has travelled (1
mile = 5280 feet)
720·3
≈ 0.409
5280
720·14
≈ 1.909
5280
720·(18.5)
≈ 2.523
5280
720·(22.2)
≈ 3.027
5280
720t
5280
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question 1
The number of calories burned while running depends on many
factors, but averages about 100 calories per mile. Suppose Nikki goes
for a run, traveling at a constant speed of 720 feet per minute and
burning 100 calories per mile that she runs.
(f) Complete the following table :
Time (in minutes) Nikki has been
Number of calories that Nikki has
running
burned
4
11
15.5
19.2
(g) Determine a function h the gives the number of calories Nikki will
burn after running for t minutes.
6
question 1 - solutions
The number of calories burned while running depends on many
factors, but averages about 100 calories per mile. Suppose Nikki goes
for a run, traveling at a constant speed of 720 feet per minute and
burning 100 calories per mile that she runs.
(f) Complete the following table :
Time (in minutes) Nikki has been
Number of calories that Nikki has
running
4
11
15.5
19.2
burned
( 720·4 )
≈ 54.545
5280
( 720·11 )
100 5280 = 150
)
(
≈ 211.364
100 720·(15.5)
( 5280 )
720·(19.2)
100
≈ 261.818
5280
100
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question 1 - solutions
The number of calories burned while running depends on many
factors, but averages about 100 calories per mile. Suppose Nikki goes
for a run, traveling at a constant speed of 720 feet per minute and
burning 100 calories per mile that she runs.
(g) Determine a function h the gives the number of calories Nikki will
burn after running for t minutes.
(
h(t) = 100
720t
5280
)
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composition of functions
The distance (in miles) Nikki travels in t minutes :
d(t) =
720t
5280
The number of calories Nikki has burned running x miles :
c(x) = 100x
The number of calories Nikki has burned running for t minutes :
t
d
d(t) =
720t
5280
h(t) = c(d(t)) = 100
c
c
( 720t )
5280
= 100
( 720t )
5280
( 720t )
5280
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composition of functions
Definition : Function Composition
Let f and g be two functions. The composition of f with g is
denoted f ◦ g and defined to be
(f ◦ g)(x) = f(g(x)).
Let f(x) = x2 and g(x) = x − 2.
Then
(f ◦ g)(x) = f(g(x)) = f(x − 2) = (x − 2)2
(g ◦ f)(x) = g(f(x)) = g(x2 ) = x2 − 2
(f ◦ f)(x) = f(f(x)) = f(x2 ) = (x2 )2 = x4
(g ◦ g)(x) = g(g(x)) = g(x − 2) = (x − 2) − 2 = x − 4
Notice that (f ◦ g)(x) ̸= (g ◦ f)(x).
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question 2
Suppose a square is growing continuously so that the length of each
side s begins with a value of 0 and grows at a constant rate of 3
inches per second.
(b) Define a function g giving the side length (in.) of the square in
terms of the number of seconds t since the square started
expanding.
(c) How does the side length of the square change from 1 second to
4 seconds ? Represent this change in side length using function
notation.
2
(d) Define a function h giving the area (in. ) of the square in terms of
the side length s (in inches) of the square.
2
(e) Define a function f giving the area (in. ) of the square in terms of
the number of seconds t since the square started expanding.
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question 2 - solutions
Suppose a square is growing continuously so that the length of each
side s begins with a value of 0 and grows at a constant rate of 3
inches per second.
(b) Define a function g giving the side length (in.) of the square in
terms of the number of seconds t since the square started
expanding.
g(t) = 3t
(c) How does the side length of the square change from 1 second to
4 seconds ? Represent this change in side length using function
notation.
g(4) − g(1) = 3(4) − 3(1) = 9
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question 2 - solutions
Suppose a square is growing continuously so that the length of each
side s begins with a value of 0 and grows at a constant rate of 3
inches per second.
2
(d) Define a function h giving the area (in. ) of the square in terms of
the side length s (in inches) of the square.
h(s) = s2
2
(e) Define a function f giving the area (in. ) of the square in terms of
the number of seconds t since the square started expanding.
f(t) = (h ◦ g)(t) = h(g(t)) = h(3t) = (3t)2 = 9t2
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question 4
Use the tables provided to answer the following questions.
x
f(x)
g(x)
−2
−1
0
1
2
3
0
3
4
−1
6
−2
5
3
2
1
−1
0
(a) g(f(−1)) = g(3) = 0
(b) f(f(3)) = f(−2) = 0
(c) g(g(0)) = g(2) = −1
(d) f(g(3)) = f(0) = 4
(e) If f(g(x)) = 3, then what is the value of x ?
If f(x) = 3, then x = −1.
Therefore, f(g(x)) = −3 implies that g(x) = −1.
If g(x) = −1, then x = 2.
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question 5
Use the tables provided to answer the following questions.
(a) f(g(2.5)) = f(1.5) ≈ 3.2
(b) g(f(4)) = g(4) ≈ 0.8
(c) g(g(3.5)) = g(1) ≈ 2.6
(d) Determine the value(s) of
x such that f(g(x)) = 2.5.
If f(x) = 2.5, then x = 1.1.
So, if f(g(x)) = 2.5, we
must have g(x) ≈ 1.1. If
g(x) = 1.1, then x ≈ 3.4.
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