Driven by Functions4.files.edl.io/88eb/04/01/20/144704-d3813bfe-5921-427e-84e6-8d4a081b7f2d.pdfOnce...
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Driven by Functions
1.1 Welcome
Welcome to “Driven by Functions,” an interactive
mathematics tutorial for students.
1.2 Objective
By the end of this tutorial you should be able to
determine if a relation between two sets of numbers
represents a function.
1.3 Prior Knowledge
In order to be successful in this tutorial you should
already know how to identify and interpret input and
output values. Recall that inputs may also be called
x-values or domain values. Outputs may also be
called y-values or range values. You should also be
able to convert a list or table of values into ordered
pairs and accurately plot those ordered pairs as
points on a coordinate plane.
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1.4 Inputs and Outputs
Congratulations! One of the nation’s newest
automobile companies has selected you to test
their most advanced self-driving car. You have
decided to take it on a road trip up and down the
coast, but before you can drive it, or rather, let it
drive you, you must understand how it functions.
When you click unlock on your car keys, what happens? The doors unlock. When you turn the key in the ignition, what
happens? The engine starts. When you press on the gas pedal, the car accelerates. When you step on the brake pedal,
the car slows down. You would expect one specific outcome to happen as a result of each of your actions. This idea, that
each time you do something, you get one result, is just like a mathematical function. When the relation between a set of
numbers is a function, we can look at each input and see that it only corresponds to one output. What about when there
are two of the same outputs? Can this still represent a function? Yes! Think about unlocking the car doors. One way of
achieving that desired outcome is to click unlock. Another way to get the doors unlocked is to use the metal key. As long
as each of the inputs only leads to one output, then we can say that the relation is a function.
When examining a relation in order to determine if it represents a function, it does not matter how many outputs each
input has. It is important to ensure that each input only has one output. There should not be multiple outputs for any
given input. Two different inputs may have the same output.
1.5 Functions
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X (domain) Y (range)
-5 -9
-1 -1
0 1
1 3
12 25
40 81
A function might be represented as a list of numbers, a set of ordered pairs, a table of values, a graph, an equation, or
even a verbal description. When you examine a set of data, you can identify whether the relation is a function by
figuring out if each individual input has one, and only one, output that goes with it. Here we can see the same function
being represented in a variety of ways. If we look at a single input, for example, when x equals negative five, we can see
that the corresponding output, or y-value, is negative nine. Notice that we can show this function as a list of inputs and
outputs, or we can even pair them as ordered pairs and write them in parenthesis like (-5,-9). These ordered pairs can
also be written in a table with inputs and outputs, and we can then plot these points on a coordinate plane. Functions
are relationships! We can take the pattern, or rule, that created the relationship and write it verbally and then as an
equation. Did you happen to see that we can take any of these inputs, double it, and add one, to get the output? That is
the rule or function for this set of inputs and outputs. We can write this function as an equation, y = 2x + 1.
1.6 Practice 1: What is a Function?
Let’s make sure that you know what a function is. Read the following choices. Then, select all of the answer choices that
would complete the following sentence to make a true statement: “When examining a relation in order to determine if
it represents a function…” Once you have made your choices, click Submit.
Sentence:
“When examining a relation in order to determine if it represents a function,”
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Correct Choice
No It does not matter how many outputs each input has.
Yes It is important to ensure that each input only has one output.
No There should be multiple outputs for any given input.
Yes Two different inputs may have the same output.
Feedback when correct:
Toot your horn! You are correct! In order for a relation to represent a function each input can only have one output.
Even when two inputs have the same output, as long as there is only one output for each input, it can still be classified
as a function. In other words, every x-value must only correspond with one y-value.
Feedback when incorrect:
Examine the correct answer choices. In order to be defined as a function, the inputs of a relation must each have only
one output. In other words, every x-value must only correspond with one y-value. If two different x-values have the
same y-values, that is okay. As long as the inputs only have one output, the relation is still a function.
1.7 Tire Trouble
Pop! Oh no, you’ve hardly begun your trip and you must have already traveled over some rusty nails! Fortunately, your
car sensed the dramatic drop in air pressure in the tires and has activated caution mode to get you safely to the nearest
tire repair shop.
When you arrive at the tire shop, the associate tells you that there will be a $49.99 inspection fee to work on such an
advanced vehicle. Luckily, the repair bill only increases $9.99 for each tire that needs to have a nail hole patched. If we
make a table of values for this situation, it will show how the repair bill is related to the number of damaged tires. When
the number of damaged tires is zero, you still have to pay the inspection fee. This point (0, 49.99) lies on the y axis. If one
X (domain) Y (range)
0 49.99
1 59.98
2 69.97
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tire is damaged, then the input, or x-value, is one, and the output, or y-value, is 59.98. A domain value of two that
corresponds to a range value of 69.97 means that the x-value represents two damaged tires, and the y-value represents
a $69.97 repair bill.
We can also describe this scenario by representing it with an equation. The total repair bill is
9.99 for each damaged tire and the $49.99 inspection fee. Knowing that the inspection costs $49.99 and each damaged
tire increases the bill by $9.99, we can take an input value and substitute it into the equation in place of the x variable.
Then we can use order of operations to determine the output, y. Let’s take the domain of 2 and substitute it in for x.
Once we solve the equation, we calculate 69.97 for the range.
1.8 Practice 2: Flat Tire Functions
With the input being the number of damaged tires, match the appropriate outputs for the possible repair bill totals in
order to complete the table when each tire costs $9.99 to fix with a set inspection fee of $49.99.
Number of damaged tires Total repair bill
0 49.99
1 59.98
2 69.97
3 Three damaged tires drop target
4 Four damaged tires drop target
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Drag Item Drop Target
$79.96 Three damaged tires
$89.95 Four damaged tires
$29.97
$39.96
Feedback when correct:
Congratulations, you’re ready to get back on the road! When represented as a table of values, a function will have the
inputs, or the domain values, on the left and the outputs, or the range values, on the right. This situation also
demonstrates a function. Each input, in this case the number of damaged tires, corresponds to one output, the cost of
the repair bill. You were able to take each input and use the function rule of y = 9.99x + 49.99 to get the corresponding
outputs.
Feedback when incorrect:
Examine the correct answer choices. When represented as a table of values, a function will have the inputs, or the
domain values, on the left and the outputs, or the range values, on the right. This situation demonstrates a function
because each input corresponds to one output. Each input, in this case the number of damaged tires, corresponds to
one output, the cost of the repair bill. It’s important that you can take each input and use the function rule of y = 9.99x +
49.99 to get the corresponding outputs.
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1.9 Cruise Control
Out on the open road again, your car works to get
as many miles per gallon as possible. To
accomplish this, the car calculates an optimum
cruising speed and then sets itself on cruise
control. You can, however, manually adjust the
speed that the car cruises at in five mile per hour
increments. When you hold your thumb on the
plus sensor built into the steering wheel, three
options appear so that you can automatically
increase the cruising speed by five, ten, or fifteen
miles per hour. When you hold your thumb on
the minus sensor, three options appear so that
you can automatically decrease the cruising
speed by five, ten, or fifteen miles per hour. You
can think of these as the inputs and the new,
adjusted cruising speed as the output.
1.10 Practice 3: Manual Override
The car is currently cruising at 70 miles per hour.
However, a new speed limit sign has recently been
installed and shows that you should only be traveling
at 65 miles per hour. What input should you use in
order to set the new cruising speed at 65 miles per
hour? Type your answer in the box and then click
submit.
Output Correct Input
75 +5
70 0
65 -5
Feedback when correct:
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Excellent, you are cruising right along! As you can see, the input does not always have to be a positive number. In a
function, the inputs and the outputs can be positive or negative.
Feedback when incorrect:
Take a look at the correct answer. If the car was cruising at seventy miles per hour but you needed to be cruising at only
sixty-five miles per hour, we would have to tell the car to decrease the speed. If we give the car an input that is negative,
we are telling the car’s computer to take away five from the speed it is traveling. As you can see, the input does not
always have to be a positive number. In a function, the inputs and the outputs can be positive or negative.
1.11 Interpreting Data and Functions
As you drive around, your car is constantly receiving and interpreting feedback from the environment. Other cars and
trucks on the road share speed, direction, and distance data so that you can avoid collisions. Your car also detects where
emergency vehicles are located. The car is programmed so that when it detects emergency flashing lights it
automatically moves out of the way and reduces its speed.
To calculate the reduced speed that the car should be traveling when emergency vehicles are preset, the car interprets
lots of data. More simply though, we can think of it like a rule that says take the current speed and reduce it by fifteen.
This sort of rule is a function because for any input, it would yield only one output. Let’s say that you were traveling at
45 miles per hour, your current speed input, and an emergency vehicle approaches with its lights on. The car would
calculate fifteen miles per hour less than the current speed and start going the new reduced speed, 30 miles per hour.
30 mph is the output of this function.
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1.12 Practice 4: Identifying Parts of a Function
We can represent the emergency vehicle function, which takes the current speed and reduces it by fifteen, as the
equation y = x - 15. Drag and drop the items to the appropriate side of the equation, assuming the original traveling
speed is 60 miles per hour.
Drag Item Drop Target
45 mph Drop target Y
New (reduced) speed Drop target Y
Output Drop target Y
Input Drop target x - 15
Current speed Drop target x - 15
Minus fifteen Drop target x - 15
60 mph Drop target x - 15
Feedback when correct:
Correct! The car uses this rule, a function, of taking the current speed, x, and reducing it by fifteen in order to get the
new reduced speed, y. The variable x is the current speed. This is also called the input. The car takes the current speed
and applies the rule, reduce the current speed by fifteen. This is shown by x minus fifteen. By taking the current speed
and reducing it by fifteen, the car generates the output, y. The output is the new reduced speed of 45 mph.
Feedback when incorrect:
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Take a look at the correct answers. The variable x is the current speed. This is also called the input. The car takes the
current speed and applies the rule reduce the current speed by fifteen. This is shown by x minus fifteen. By taking the
current speed and reducing it by fifteen, the car generates the output, y. The output is the new, reduced speed of 45
mph.
1.13 Time to Reflect
You made it to the coast! As the sun is setting and your trip comes to an end, you get a chance to reflect on what it
means to be a function. You begin to realize that many of life’s every-day experiences demonstrate the variety of
different ways that situations can be represented by functions. A list of numbers, a set of ordered pairs, a table of
values, a graph, an equation, or a verbal description can all represent a function as long as each input has only one
output. You tell the car to drive you home. After it drops you off at your house, it returns to the dealership.
1.14 To Buy or Not to Buy?
It has been one month since you sent back the test vehicle. While checking your social media feeds, you notice a
message from the dealership. Because you gave the car two thumbs up in your review post, they have sent you an offer
to let you purchase the car at an extremely discounted price of only $9,499! You reply and tell them that you are
absolutely interested and would like some more information. They provide a graph to explain how purchasing would
work. Since your review was so positive for their company, the dealership is willing to pay the first payment of five
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hundred dollars. Your monthly payments will be five hundred dollars until you have paid off the balance. As you look at
each point on the graph, you realize that this is another example of a functional relationship. When you first get the car
at month zero, the balance is almost nine thousand dollars. After one month, you would owe about eighty-five hundred
dollars. The remaining balance is decreased by five hundred dollars every month. Each input has one output.
1.15 Practice 5: Buying the Car
In the fine print of the offer, you realize that the dealership says the graph represents approximate values and is to be
used for illustrative purposes only. To calculate the exact remaining balance each month, you must use an equation, y = -
500x + 8,999. Using the function rule, calculate the exact value of the remaining balance after owning the car for 6
months. Type your answer in the space provided and then click submit.
Months of Ownership Correct Remaining Balance
0 8999
6 5999
12 2999
18 0
Feedback when correct:
You got it! When x equals six, y equals five thousand nine hundred ninety-nine. So after six months of ownership, the
remaining balance would be $5,999. To calculate the remaining balance after six months, you substitute six into the
equation for the variable x. Negative five hundred times six would be negative three thousand. Negative three thousand
plus eight thousand nine hundred ninety-nine equals five thousand nine hundred ninety-nine. So after six months of
ownership the remaining balance would be $5,999.
Feedback when incorrect:
Look at the correct answer. We need to know the y-value, or the output, when x, the input, is equal to six. To calculate
the remaining balance after six months, you substitute six into the equation for the variable x. Negative five hundred
times six would be negative three thousand. Negative three thousand plus eight thousand nine hundred ninety-nine
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equals five thousand nine hundred ninety-nine. So after six months of ownership, the remaining balance would be
$5,999.
1.16 Practice 6: Not a Function?
Let’s keep practicing. Which of the following does not represent a function? Select your answer and click submit.
Correct Choice
No Choice A: the data points (-2.5,4), (-1,9), (0,0), (1,10), (10,3), (12,5) and a graph of those
points
Yes Choice B: the data points (-5,-9), (-1,-1), (-1,2), (1,3), (12,25), (10,10) and a graph of those
points
No Choice C: a table with a colum of x values that reads -0.75, -0.5, 0, 1.1 and a column of y
values that reads -11, 0, 1.1, 2, plus a graph of those values
No Choice D: a table with a column of x values that reads -21, -6, 12, 18 and a column of y
values that reads 21, 3, -5, -30, plus a graph of those points
Feedback when correct:
Wonderful! You could see that in answer choice B, there were two outputs for the input of negative one, and you know
that a function has to have only one output for each input.
Therefore, this relation does not represent a function.
Feedback when incorrect:
Look at the correct answer. Remember that to be a function there can only be one output for any given input. In the
correct answer choice, when the input, or x-value, is negative one, there are two outputs, or y-values, both negative one
and two. This can not be a function because there is not only one output for each input.
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1.17 Practice 7: Matching Functions
One more! Drag and drop each function so that it is matched with an equivalent representation of the function.
Drag Item Drop Target
(-9,12), (-3,16), (0,20), (3,24), (9,28) Inputs: -9, -3, 0, 3, 9
Outputs: 12, 16, 20, 24, 28
Take thirty away from half of a number. Y = 0.5x - 30
T chart: X values: -5, -1, 0, 1
Y values: -9, -1, 1, 3
Graph
Feedback when correct:
Fantastic! You can tell that functions can be represented in a variety of ways, all of which show that each input has one
output. The given list of inputs and outputs corresponds to the given list of ordered pairs. The equation y = 0.5x – 30 can
be verbally represented by saying take thirty away from half of a number. When plotted as points on the coordinate
plane, the ordered pairs in the table of values can each be identified on the graphed line.
Feedback when incorrect:
Look carefully at the correct answers. Notice that each of the same functions can be represented in different ways. The
inputs and outputs still match, regardless of how the function is represented. The given list of inputs and outputs
corresponds to the given list of ordered pairs. When the input was negative nine, the output was 12. That corresponds
to the ordered pair (-9,12). In the equation y = 0.5x – 30, you have some number, x, being multiplied by 0.5, which is the
same thing as taking half of the number. Since you subtract 30 from that product you can verbally represent the
equation by saying take thirty away from half of a number. When plotted as points on the coordinate plane, the ordered
pairs in the table of values can each be identified on the graphed line.
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1.18 Lesson Review
In this tutorial, we drove home the concept of a function. You learned how to determine if a relation represents a
function. You learned that when a relation represents a function, it means that each input has exactly one output. You
also learned that sometimes an output might have more than one input, and as long as the input only has one output,
the relation is still considered a function. Input and output values can be positive or negative. Finally, you also learned
that functions can be represented as lists of numbers, ordered pairs, tables of values, graphs, equations, and verbal
descriptions.
1.19 Thank You
Thank you for using this original tutorial. Be sure to check out our other original tutorials too.
---------------------------------
Credits
All images licensed from Getty Images, iStock.com and/or Thinkstock.com, unless otherwise noted.
“Tire Puncture Pop Hit Punch” by JohnsonBrandEditing, CC0, Freesound,
https://freesound.org/people/JohnsonBrandEditing/sounds/243375/
Credits (Slide Layer)