STEM 698 Algebra Initiative
Homework assignment due Tuesday, 4/9
1. CME Algebra page 424, Question 5
a. Take the opposite of the input and then add 2
A function
Input Output
0
2
1
1
2
0
3
−1
4
−2
b. Square the input and then subtract 4
A function
Input
0
1
2
3
4
Output
−4
−3
0
5
12
x 2  3x
This assignment is not a function because division by 0 is undefined.
xx
d. x produces a number that is 4 units to the left of x on the number line
c.
x
A function
Input Output
0
−4
1
−3
2
−2
3
−3
4
−4
e. x produces a number that is 4 units away from x on the number number line
This assignment is not function because there are two outputs for each input (e.g. 0
maps to 4 and −4).
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2. CME Algebra page 424, Question 6
a. The input is a day of the year. The output is the average temperature in Barcelona on
that day.
This assignment is a function. There is only one average temperature for each day.
b. The input is the speed of the car. The output is the time it takes for a car moving
constantly at that speed to travel 100 miles.
This assignment is technically not a function. For every non-zero speed there is only
one such time. However, if the speed is zero, then no such time exists. If one
restricts the domain to be non-zero speeds, then it is a function.
c. The input is a positive number. The output is a number who absolute value is the
input.
This assignment is not a function. For instance
5 has two possible outputs, both
5 and  5 .
d. The input is a year. The output is the population of the United States during that year.
This assignment is not a function, because the population of the United States varies
during the year.
3. CME Algebra page 434, Question 9
a. The input is a letter. The output is any word starting with that letter.
This assignment is not a function. E.g. “a” maps to “act” and “algebra.”
b. The input is a person. The output is that person’s age in years.
This assignment is a function.
c. The input is the name of a month. The output is the number of days in that month.
This assignment is not a function because February has 28 days in non-leap years and
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29 days in leap years.
4. CME Algebra page 430, Question 9
g ( x)  x 2  5 and h( x)  | 3x  1|
a.
g (0)  5, g (1)  4, g (1)  4
b. h(0)  1, g (1)  4, g (1)  2
c.
g (3)  5  9
d. g (4)  h(4)  24
5. CME Algebra p. 434 Problem 12 a - f , k, and m.
f ( x)  3x  2
a.
g:x
x2
3
g ( x) 
3
x2
f (7)  23
b. g (7)  53
c.
g (7)  53
d.
f ( g (5))  f (1)  5
e.
g ( f (5))  g (17)  5
f.
h( g (5))  h(1)  3
k. f (a  2)  3a  8
m. f ( g ( x))  f ( x3 2 )  3( x3 2 )  2  x
6. CME Algebra page 439, Question 2
The first two graphs are not the graphs of functions (they violate the vertical line test),
but the second two are.
7. CME Algebra page 445, Questions 1, 2, 3, 4
C, B, C D
8. Rent-A-Reck Truck Rental offers a daily truck rental of $20 plus $0.25 per mile driven
that day. U-Drive-It Truck Rental offers a comparable sized truck at rate of $25 plus
$0.15 per mile driven that day.
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a. Let R(x) denote the function whose input is miles driven and whose output is the cost
of renting a truck from Rent-A-Reck. Give an algebraic expression for R(x).
R(x) = 20 + 0.25x dollars
b. Let U(x) denote the function whose input is miles driven and whose output is the cost
of renting a truck from U-Drive-It. Give an algebraic expression for U(x).
U(x) = 25 + 0.15x dollars
c. Enter both of these functions into your calculator. Using the table feature determine
under what conditions it is cheaper to rent from Rent-A-Reck and under what
conditions it is cheaper to rent from U-Drive-It.
Your table should show that it is cheaper to use Rent-A-Reck if you are going to drive
less than 50 miles. If you are going to drive more than 50 miles, it is cheaper to rent
from U-Drive-It.
d. Repeat part c using the graphing feature of the calculator. Sketch your graph on the
coordinate axes below (or draw your own your own coordinate axes). Explain how
the graph shows under what conditions it is cheaper to rent from Rent-A-Reck and
under what conditions it is cheaper to rent from U-Drive-It.
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e. Algebraically, determine under what conditions it is cheaper to rent from Rent-AReck and under what conditions it is cheaper to rent from U-Drive-It.
Since the graphs are lines, we need to find where R(x) = U(x).
20  0.25 x  25  0.15 x
0.10 x  5
x  50
f. Briefly explain how the above problem exemplifies the “Rule of Four” that Lynn told
you about.
The problem uses words, tables, graphs, and formulas
9. Astronauts looking at Earth from a spacecraft can see only a portion of the surface.
The fraction F of the surface of Earth that is visible at a height h, in kilometers, above the surface
is given by the function
F ( h) 
0.5h
6380  h
a. Express the fraction of the earth visible from 22000 kilometers in functional notation and
evaluate.
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F (22000) 
0.5  22000
50

 0.388
6380  22000 129
b. At what height is a fraction of 0.40 (i.e. 40%) of the Earth’s surface visible? Briefly
explain the approach you took to get your answer. (Hint.You can use algebra or use a
table or even a graph…)
We need to solve 0.4 
0.5h
.
6380  h
Here is the algebraic approach:
0.5h
6380  h
0.4(6380  h)  0.5h
2552  0.4h  0.5h
2552  0.1h
25520  h
0.4 
At 25520 miles, the fraction reaches 0.4.
Here is a graphing calculator approach:
Click on “Y=”
In Y1 type “0.5*X/(6380+X)”
In Y2 type “0.4”
Now set your window:
Xmin =0
Xmax =40000
Ymin=0
Y max =0.7
Click Graph.
Now click “2nd Calc-> Intersect”
And find the intersection of 0.4 and Y1.
Here is a screen shot.
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