Domain and Range

Draw

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It starts at (-1, 4) and then it includes every x-value to the right thereafter.​

Therefore, the domain is ALL X VALUES greater than or equal to -1.

We write this as: {x|x≥−1}

This is a relation

It starts at (-1, 4) and then it includes every y-value below it.​

Therefore, the range, is ALL Y VALUES less than or equal to 4.

We write this as: {y|y≤4}

This is a relation cont...

Poll

How does this feel so far?

• I get it!

• Can we look at that again?

All the inputs of a function are the domain of the function. You can put fruit in a blender.
Fruit is in the domain of the blender.

You cannot put a car into a blender.
A car cannot be in the domain of the blender.

​

inputs = domain

The outputs of a function are the range of the function.
A blender can make a smoothie.
A smoothie is in the range of the blender. 

A blender cannot make a book. A book is not in the range of the blender.

​

outputs = range

Open Ended

If the oven is our function, what is a possible domain/input? What would be the corresponding range/output?

the range!

​​What do we call the set of all outputs for a fuction?

the domain!

​What do we call the set of all inputs for a fuction?

You may see a domain or range in this form:
the set of all ___
A set is a group, or collection of things. In algebra, a set is usually numbers.

actual
realistic
applied
real world
in a context

​Practical

allowable
possible
not applied
not real world
no context

Theoretical

Theoretical vs Practical Domains and Ranges

Multiple Choice

Which type of domain and range should you use for real-world problems?

• Theoretical

• Practical

Categorize

Options (10)

allowable

possible

not applied

not real world

no context

actual

realistic

applied

real world

in a context

Organize these options into the right categories

Theoretical

Practical

• all possible outputs of a function

• based on what is allowable in math

​Theoretical range

• all possible inputs of a function

• based on what is allowable in math
  

Theoretical domain

Theoretical

• all actual outputs of a function

• based on applying the function to a real-world situation

Practical range

• all actual inputs of a function

• based on applying the function to a real-world situation
  

Practical domain

Practical

Poll

Checking in! How are we feeling now?

• I'm doing well!

• I'm a little confused, but will read the lesson to clarify

• I'm really confused and will read the lesson for clarification

• I'm lost. I'll reach out for a 1-1 after i've read the lesson.

Let's put all this into action.

A vendor has 20 umbrellas to sell. She sells them for $15 each.

The function m(u) = 15u models the total amount of money the vendor makes from selling u umbrellas.

​Determine the theoretical domain and range:
What are the possible inputs and outputs? Any real #

Determine the practical domain and range:
What are the realistic inputs and outputs? 0-20; 0-300

m(u)=15u

domain (u) with any real number

there are an infinite number of ordered pairs

Graph of function with theoretical domain

m(u)=15u

domain (u) with inputs of 0-20

the domain is set to 21numbers, so there are 21 ordered pairs

Graph of function with practical domain

Mr. Cruz will buy more than 0 gal of gas. If his tank was full, he would not be buying gas.

If Mr. Cruz’s tank is empty (which means his car was towed to the gas station) then he would purchase 15.5 gal of gas.

Mr. Cruz can buy any fraction of a gallon of gas.

What are the least and greatest amounts of gas Mr. Cruz will buy?

The practical domain is the set of all real numbers greater than 0 and less than or equal to 15.5.

Determine the practical domain

Mr. Cruz is filling his gas tank at a station where gas is selling for $2.48/gal. He has a coupon for $5 off his purchase. The function p(g)=2.48g−5 represents the amount Mr. Cruz will pay for g gallons of gas. Mr. Cruz’s gas tank has a capacity of 15.5 gal. What is the practical domain and range of the function?

For some input values, the output is negative. For example, p(1)=−2.52. But the gas station will not pay Mr. Cruz for taking a gallon of their gas. Because of his coupon, the least amount Mr. Cruz will pay is $0.

Find p(15.5) to find his greatest possible cost.

What are the least and greatest amounts of money Mr Cruz will pay?

p(g)=2.48g−5

p(15.5)=2.48⋅15.5−5

 =33.44

The practical range is set of all real numbers between 0 and 33.44, inclusive.

Determine the practical range

Mr. Cruz is filling his gas tank at a station where gas is selling for $2.48/gal. He has a coupon for $5 off his purchase. The function p(g)=2.48g−5 represents the amount Mr. Cruz will pay for g gallons of gas. Mr. Cruz’s gas tank has a capacity of 15.5 gal. What is the practical domain and range of the function?

Fill in the Blanks

The owner of a tree farm has 10 evergreen trees left to sell. Each one sells for $40. The function r(t)=40t models the revenue the owner can make by selling the t remaining trees. What is the least number of trees that can be sold?

Fill in the Blanks

The owner of a tree farm has 10 evergreen trees left to sell. Each one sells for $40. The function r(t)=40t models the revenue the owner can make by selling the t remaining trees. What are the greatest number of trees that can be sold?

Fill in the Blanks

The owner of a tree farm has 10 evergreen trees left to sell. Each one sells for $40. The function r(t)=40t models the revenue the owner can make by selling the t remaining trees.

Given the practical domain is all whole numbers 0-10, what is the least amount the owner can make?

(substitute 0 for t)

Fill in the Blanks

The owner of a tree farm has 10 evergreen trees left to sell. Each one sells for $40. The function r(t)=40t models the revenue the owner can make by selling the t remaining trees.

Given the practical domain is all whole numbers 0-10, what is the greatest amount the owner can make?

(substitute 10 for t)

Multiple Choice

The owner of a tree farm has 10 evergreen trees left to sell. Each one sells for $40. The function r(t)=40t models the revenue the owner can make by selling the t remaining trees.

What is the practical range of the function?

• all multiples of 40 between 0 and 400, inclusive 

• all real numbers between 0 and 400, inclusive

• all multiples of 10 between 0 and 40, inclusive

Open Ended

Self-reflect: How did you do? What do you think would help you to understand better? What could I do as your teacher to help you?