STEM 698 Algebra Initiative
Homework assignment due Tuesday, 4/23
1. Consider the piecewise defined function
for x  1
 13 x  2
 1
f ( x)   2 x  1 for 1  x  5
 3 x  9 for x  5
2
whose graph is shown below:
a. What is f (3) ?
b.
c.
d.
e.
f.
1
3
(3)  2  1
What is f (1) ? 13 (1)  2  73
What is f (5) ? undefined
For what values of x (if any) does f ( x)  0 ? −6 ,2, and 6
For what values of x (if any) does f ( x)  1 ? −3 and 203
For what values of x (if any) does f ( x)  4 ? −18
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2. Find an expression for f ( x) . Assume that all marked points have integer coordinates
and assume that if a line seems to pass through a point with integer coefficients then it
does.
 2 x  10 for x  1
3
3
f ( x)  
 5 x  26 for x  1
3
 3
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3. CME Algebra I Page 445 Question 8 (I left you some space).
Function N ( d ) tells you the number of candles you can buy with d dollars. Each candle
costs $3.
a. Calculate N (25 12 ) and N (308) . N (25 12 )  8 candles; N (308)  102 candles.
b. Find two values of d such that N (d )  70 . For example N (210)  70 and N (211)  70 .
c. Graph N ( d ) for inputs of d between 0 and 10.
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4. CME Algebra I Question 7 on page 400.
One taxi charges $1.10 for the first mile and $1 for each additional mile. Another taxi charges
$2.30 for the first mile and $0.60 for each additional mile. For what distances will the first taxi
be less expensive than the second?
Let x be the number of miles driven. The cost (in dollars) for the first taxi as a function of
miles can be given by:
1.1 for 0  x  1
f ( x)  
for x  1
1.1  ( x  1)
The cost (in dollars) for the second taxi as a function of miles can be given by:
2.3 for 0  x  1
f ( x)  
for x  1
2.3  0.6( x  1)
Below is a graph for each function.
Solving 1.1  ( x  1)  2.3  0.6( x  1) , you will get x = 4. So your cab ride is less than 4 miles,
then first taxi costs less; if is more than 4 miles, then the second taxi costs less.
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5. (CME Algebra I, page 402, Question 6) Scott leaves home and walks 1.5 miles to school at a
rate of 3 miles per hour. Fifteen minutes later, his brother leaves for school. He is riding his
bike at a rate of 9 miles per hour.
a. Draw a distance-time graph to represent this situation.
Here are two graphs you could make. The first one has the x-axis in hours.
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In this case the equation for Scott’s distance from home in miles is f (t )  3t . The equation for
Scott’s brother’s distance from home in miles is
0 for 0  t  14
f (t )  
1
1
9(t  4 ) for t  4
Alternatively, you can use minutes on the x-axis:
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In this case the equation for Scott’s distance from home in miles is f (t )  603 t . The equation for
Scott’s brother’s distance from home in miles is
0 for 0  t  15
f (t )   9
 60 (t  15) for t  15
b. When will Scott’s brother overtake him? (Find the answer algebraically and check it on
the graph above.)
E.g. solving 9(t  14 )  3t yields t = 3/8 of an hour or 3/8*60 = 22.5 minutes.
c. How far from home will they be when they meet?
3(3/8) = 9/8 = 1.125 miles.
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6. (Review problem) Find all values of x that satisfy the inequality | x  9 | 1  x . A graphing
calculator could come handy, but there are other ways to solve this problem as well. Show your
work.
Graphically one would graph both f ( x) | x  9 | 1 and g ( x )  x :
From the graph you can see that the function in red is below the function in blue when x  4
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7. Suppose f is a function.
a. If 10  f ( 4) , give the coordinates of a point on the graph of f.
(−4,10)
b. If 6 is a solution of the equation f ( w)  1 , give a point on the graph of f. (6,1)
8. A downtown city parking lot charges $1.50 for each 30 minutes you park, or fraction
thereof, up to a maximum of three hours. This means that if you park for 3 minutes or
10 minutes or 30 minutes , you pay the same amount , namely $1.50. If you park 31
minutes or 49 minutes you pay $3.00 etc. Let C be the function that assigns to each
length of time you park t (in hours), the cost of parking in the lot C (t ) in dollars.
a. Complete the table below:
t in hours
0
1
4
1
3
9
16
11
2
29
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C (t ) in dollars
$0 or $1.50 depending on how you
interpret “parking for 0 minutes”
$1.50
$1.50
$3.00
$4.50
$7.50
b. Sketch a graph of C (t ) for 0  t  3 hours.
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c. What kind of function is C (t ) ? C (t ) is a step function.
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