Driven by Functions4.files.edl.io/88eb/04/01/20/144704-d3813bfe-5921-427e-84e6-8d4a081b7f2d.pdfOnce we solve the equation, we calculate 69.97 for the range. 1.8 Practice 2: Flat Tire

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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.


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


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,”


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


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


Drag Item Drop Target

$79.96 Three damaged tires

$89.95 Four damaged tires

$29.97

$39.96


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.


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.


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



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.


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.


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.


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


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.



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


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


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.


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


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


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.


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.


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.


(-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


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.


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.


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)

Documents

Driven by Functions4.files.edl.io/88eb/04/01/20/144704-d3813bfe-5921-427e-84e6-8d4a081b7f2d.pdfOnce we solve the equation, we calculate 69.97 for the range. 1.8 Practice 2: Flat Tire