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STEM 698: Homework Assignment Due Tuesday, June 4 (Last assignment of
quarter) Solutions
2. A parking garage charges $6 for the first hour (or fraction thereof) and then $2 for
each half an hour (or fraction thereof) up to a daily maximum of $30. Let C be the
function whose input is time parked (in hours) and who output is parking cost (in
dollars).
a. Fill in the following table:
t in hours
Cost in
dollars
D
1 14
Any value
x where
4.5  x  5
in hours
3
31
12
1
$8
$22
$14
$14
b. Sketch a graph of the parking cost (in dollars) as a function of time parked (in
hours) for 0  t  4 hours
3. (Adapted from a published Teacher Qualifying Exam) Consider the following two
cellular plans:
Plan A (Standard plan) A basic fee $30 for the first 500 minutes plus $0.60 per minute
for every minute over.
Plan B (Pay as you go) $0.25 a minute with no monthly fee.
(Note that both plans use technology that allows them to charge for the exact amount of
time used; they do not “round up” the time to the nearest minute.)
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a. On the graph below, plot total cost of each plan as a function of minutes called. The xaxis should be from 0 to 1200 minutes.
b. Express the cost of each plan in function notation (x denotes the number of minutes
used)
30 for 0  x  500
Plan A: f ( x)  
30  0.6( x  500) for x  500
Plan B: f ( x)  0.25 x
c. For how many minutes is the cost for the two plans equal? (There are two minute
amounts for which they are equal.)
The two plans are the same cost for 120 minutes and for 5400 / 7 or 771 73 minutes
or about 771.43 minutes.
d. Write a short paragraph comparing the two plans. In particular, address the
considerations that would be helpful for someone making a decision about which plan
to use.
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Plan A is cheaper if you use less than 120 or more than 771 73 minutes. Plan B is
cheaper if you used between 120 and 771 73 minutes. In practice, Plan B is probably
only useful for someone who makes very few calls.
4. (Adapted from a published Teacher Qualifying Exam) Martin and Fidel are
running a race. Martin starts out at a speed of 2 meters per second, much faster than
Fidel who starts out at 1.4 meters per second. However, after 40 seconds, Martin
slows down to a speed of 1.2 meters per second. Fidel continues to maintain a steady
pace of 1.4 meters per second.
a. On the coordinate system below, sketch a graph of both Martin’s distance and Fidel’s
distance as a function of time for the first 200 seconds.
Distance
from the
start of the
race (meters)
Time (seconds)
b. Express the Martin and Fidel’s distance from the start as a function of time in seconds
in function notation.
2 x for 0  x  40
Martin: f ( x)  
80  1.2( x  40) for x  40
Fidel: f ( x)  1.4 x
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b. At what distance does Fidel overtake Martin? 224 meters
5. (Adapted from CME Algebra I page 358 Question 5).
Conan waits at the train station for Courtney. Courtney gets off the train 217 feet
away from where Conan is waiting on the platform. They start running toward each
other. Conan runs 11 feet per second. Courtney runs 10 feet per second.
a. Make a graph of this situation for 15 seconds, with time on the x-axis and
distance from where Conan was waiting is on the y-axis.
b. How long will take for them to meet? 10 13 seconds
c. How far will Conan have run when they meet? 113 23 feet
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6. Decide whether each description represents a function. Explain.
a. The input is a page number of a particular book. The output is the number of
words on the page.
This relationship is a function. For each page there is only one number that is the
count of words on that page.
b. The input is the area of a square. The output is the perimeter of the square.
This relationship is a function. If the area of a square is x, then the side length of
the square must be
x and so the perimeter must be 4 x . So given the area x,
there is only possible value for the perimeter, namely 4 x .
c. The input is the area of a right triangle. The output is the perimeter of the
triangle.
This relationship is not a function. For example consider a right triangle with side
lengths 3,4, and 5 and another right triangle with side lengths 1, 12, and 13 .
The area in each case is 12 square units, but the perimeter in the first case is 12
units and the perimeter in the second cases is 13  13 units. So the input 12 is
associated with at least two different outputs.
d. The input is the circumference of a circle. The output is the area of the circle.
This relationship is a function. If the circumference is C, then the radius must be
 
2
2
C . Hence the area is
π C  C . The circumference of a circle determines
2π
2π
4π
the radius of the circle which in turn determines the area. So there is only one
possible output for any given input.
e. The input is the coordinate pair representing a point in the plane (x, y). The
output is the sum of the coordinates.
This relationship is a function. There is only one sum for any two real numbers.
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f. The input is a real number. The output is the greatest integer less than or equal to
x.
This relationship is a function. While there are many integers less than a
particular real number, there is only one that is largest.
g. The input is a real number. The output is an integer less than or equal to x.
This relationship is a function. There are many integers that are less than a
particular real number. For example both 1 and 2 are less than 2.4. So both 1
and 2 would be outputs for the input 2.4.
h.
2 x for x  3
f ( x)   2
8  3 x for x  3
This formula defines a function. For x  3 and x  3 , the formulas give one
possible output for any given input. For x = 3, there are two formulas. But since
2  3  6 and 8  23  3  6 , there is only one output at x = 3 as well. Below is the
graph of f ( x) .
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i.
5  2 x for x  1
f ( x)   2
3x for x  1
This formula does not define a function because there are two possible outputs
for x = 1 , namely 5  2(1)  7 and 3(1)2  3 .
One can see this from the graph:
The graph violates the vertical line test at x = −1.
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7. Find the domain of the following functions (here we are following the “domain
convention” whereby the domain of a function given by a formula is the largest
set of real numbers for which the formula as written is mathematically
meaningful).
a.
x2  4
f ( x) 
x2
All real numbers x except 2.
b.
f ( x) 
7x
x2
All real numbers x greater than 2.
\
c.
f ( x)  | x  3|
All real numbers
d.
f ( x) 
x3
x
All real numbers greater than equal to −3 except 0.
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