Applications of Domain and Range

Applications of Parent functions

​So, yesterday we looked into parent functions their attributes, such as, domain, range, x & y intercepts, and asymptotes. We also gave a peek into the application of those functions introducing the concept of "restrictions."

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*Therefore, today we will further look into the application and restrictions of functions. We will mainly use those restrictions to understand the domain and range of the situation while using coordinates for results or answer to questions. ​

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**Todays success criteria is to effectively apply our knowledge of the attributes of parent functions to describe or answer questions about the given situation.​

​Quick Review

Functions are one-to-one relationships, this means that for each x there is one y. To further simplify since x is the independent it cannot repeat, however​ y being the dependent is able to repeat.

When looking into a table: When looking into a graph:

​x	​y
​1	​8
2​	9​
3​	10​

​If x doesn't repeat then it is a function.

​If the graph passes the vertical line test then it is a function.

​Some functions are natural restricted, for example the quadratic function:

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​If you notice the "curve", actual name "vertex" you can see that this graph will never reach the negatives, when talking about range.

​Therefore, when talking about the quadratic parent function. The range is naturally restricted to zero and all positive numbers greater than zero

​Moreover, when applying functions to real-life situations we encounter all types of restrictions/boundaries

​Alright! hopefully that short review helped you get back on track.

Now, lets look at a couple of examples:

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Joe had a summer job that pays $7.00 an hour and he worked between 15 and 35 hours every week. His weekly salary can be modeled by the equation: S = 7h, where S is his weekly salary and h is the number of hours he worked in a week.

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1. Describe the domain and range for this problem using appropriate notation.

2. What does the order pair (20,140) mean in this problem?​

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​Let's look at the important information:

• ​Job pays $7.00 per hour

• He worked between 15 and 35 hours

• His function is S = 7h → means Salary = $7 times (hours)​

• So the amount of money "S" depends on the amount of "hours"

• Domain = independent / Range = dependent

  Now first question​

​1. Describe the domain and range for this problem using appropriate notation.

Hours = independent. So since he worked between ​15 and 35 hours.

The are two ways to properly write this. They are both called interval notation

• 15 ≤ x ≤ 35 or [15,35]​ → they both mean the same "he worked between 15 and 35 hours.

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​Now, onto our next question:

2. ​What does the order pair (20,140) mean in this problem?

​Well, lets examine:

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​This is a picture of the graph already restricted on the domain.

​If domain is the hours worked and range is money, then (20,140) means that he worked for 20 hours and earned $140

​Alright! next example:

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​The event center at the Turning Stone resort has a seating capacity of 5,000  seats.  The amount of money brought in by an entertainment event, M, is a  function of the number of people, n, in attendance.  Each ticket costs $55. 

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​ 1. Write a function to model the money brought in, M, in relation to the people, n, in attendance.

2. ​What is the domain of this function (in context)?

3. ​What is the range of this function (in context)?

​Lets get the important stuff:

1. ​Sitting capacity of 5,000 people

2. ​Money, M, is a function of people, n, in attendance.

3. ​Each ticket costs $55.00

​Now, lets look at the first question:

Write a function to model the money brought in, M, in relation to the people, n, in attendance.​

"People in attendance" are people that paid so every person in attendance paid the $55.00. So the equations should be

M = 55(n) ​

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​Ok, onto the next question:

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2. What is the domain of this function (in context)?

ok, so the amount of money depends on the amount of people in attendance. So, money is the dependent and people in attendance is the domain.

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Therefore, the domain (in context) are the amount of people in attendance.

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​What is the range of this function (in context)?

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Following the last question: The range is the total amount of money collected by selling each ticket for $55.00​

​Alright! Let's have you all try a few questions.

Multiple Choice

Leslie's car travels about 25 miles per gallon of gas. Her car needs 14 gallons of gas to be full. What is a reasonable domain?

• $$x\ge25$$

• $$0\le y\le14$$

• $$x\le14$$

• $$0\le x\le14$$

Multiple Choice

For the function (0,1), (1,-3), (2,-4), & (-4,1), which choice gives the correct domain & range?

• D: {0, 1, 2, -4}
  R: {1, -3, -4, 1}
• D: {-4, 0, 1, 2}
  R: {-4, -3, 1}
• D: {0, 1, 2, 3, 4}
  R: {1, -3, -4}
• D:  {-4, -3, 1}
  R: {-4, 0, 1, 2}

Multiple Choice

Mario has and iTunes gift card worth $20.00. He uses the formula    y=0.99x to determine y, the total cost for x number of songs. What is the domain for the situation?

• All integers between 0 and 20
• All real numbers between 0 and 19.80
• All integers greater than 0
• All integers between 0 and 250

​Now that we have explore restrictions and how the affect the domain and range of a graph, we can move onto a little more challenging examples:

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The formula for throwing a baseball in the air is represented by h = – 16t² + 12t + 40 where h is the height of the ball. After how many seconds will the ball hit the ground?

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So lets get the important information out:

• h = the height of the ball​

• ​The questions says "after how many seconds..."

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• ​Then they give us the formula h = – 16t² + 12t + 40

• So h = height and t = seconds (time)

• ​So, the height of the ball depends on the time

​Now, the question: "After how many seconds will the ball hit the ground?"

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​As long as we understand that domain is the time and range is the height. We can type our equation in our fancy desmos graphing calculator or classroom graphing calculator (they are on the first drawer in the file cabinet) and get an answer

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Next slide I'll show a picture of the desmos graping. BTW you can google desmos graphing and you can use it for free! ☻

​Again, Domain is time and range is height

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So, (0,40) is the initial height "the ball has not been thrown.

Then (2,0) two seconds pass and the ball is at 0 height, meaning the floor. ​

​To answer the question: After how many seconds will the ball hit the ground? That would be 2 seconds

​Understanding Domain and Range give us the foundational skill to understand restrictions and limitations. It also allows us to see things a bit different, for example:

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You are supposed to meet your friend at your house. You call your friend because he/she is running late and they say " I'll be there in less than five minutes".

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What is the domain of the function? *Given that the amount of distance cover is always dependent on time

Then domain is: 0 ≤ x < 5 ​

0 = the moment you hung up and 5 is the limit that your friend gave.​

​If you understand domain and range, you understand most problems. Take a look at the next one:

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​A company has developed a new video game console. After completing cost analysis and demand forecasts, the company has determined that the profit function for the new console is  f(g) =250g²+70,000g-4,570,000  where g is the number of consoles sold. What is the domain of the profit function?

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Ultimately it boils down to: Money = number of consoles sold​

so D: g ≥ 0 (because there is the possibility that they don't sell any) (But also the possibility that they sell billions) the upper limit is not given.​

​Alright! I'll give you a few more to try and then you can move onto step 3 your independent practice

​Hopefully you did great on this questions! Onto Step 3! YAY!

Multiple Choice

The graph of quadratic function f is shown on the grid. Which of these best represents the range of f?

• All real numbers less than -8 or greater than 0.

• $$-2<x\le7$$  

• $$-8\le y\le0$$  

• All real numbers

Multiple Choice

What is the domain of the function for this situation?

• 0<h<80
• 0≤h≤80
• 0≤t≤4.5
• 0<t<4.5

Multiple Choice

The height of a basketball after it bounces is represented by h(t) = -8t² + 50t where h represents the height, in feet, and t represents the number of seconds that have passed since the ball bounced off the ground.

What is the input?

• Height in feet

• Height in inches

• Number of seconds that have passed

• Number of basketballs

Multiple Choice

The height of a basketball after it bounces is represented by h(t) = -8t² + 50t where h represents the height, in feet, and t represents the number of seconds that have passed since the ball bounced off the ground.

What is the output?

• Height in feet

• Height in inches

• Number of seconds that have passed

• Number of basketball

Multiple Choice

George is driving from Orlando to Jacksonville. The distance between the two cities is approximately 250 miles. His average speed is 55 miles per hour. The function that represents how many miles George has left on his trip after z hours is f(z) = 250 – 55z

What is the input?

• Cars

• Miles left to travel

• Hours

• Roads

Multiple Choice

A pizza artist puts a pizza in the oven for 30 minutes. Let P(x) = 0.5x² + 65 represent the temperature of the pizza x minutes after you put it in the oven.

P(0) = 65 means that….

• The pizza was at 0 degrees after 65 minutes

• The pizza was 65 degrees at 65 minutes

• The pizza was at 65 degrees at 0 minutes

• There were 0 minutes and 65 pizzas