Class Plan

Hello class! I am professor Cavano.  Today I want to teach you about Domain and Range.  Domain and range are especially important when talking about functions.  For many functions, numbers beyond a certain point or just an assortment of numbers are not effective and act as a boundary or cutting off point for a function. These numbers are like milk gone bad – they expire at certain time or number and no longer work.  You could think about domain and range as a fence that holds a dog in from escaping its yard.  If that fence breaks, the eager dog escapes and all is lost in the world.  Similarly, if these numbers are used, all is lost and the function does not work.  After this class, I want you to be able to identify the domain and range for a function in both a real life and mathematical frame of mind. 

Domain is the set of all input values that correspond to specific output values. Range is the set of all the output values. Range relies on Domain. You could think of domain as the x values and range as the y values

To figure out domain and range, we must first classify the three types of forms of writing for the domain and range.  The first one is called set builder notation.

Set builder has brackets that are shaped like { } and has interesting symbols that represent different numbers.  The symbols are as follow

·      Natural Numbers N - {1, 2, 3, 4, 5,…}

·      Rational Numbers – Q { -1/2, 0, 1/2}

·      Irrational Numbers – p {Pi or e}

·      Real Numbers R { natural, rational, irrational numbers } (not included: imaginary numbers, such as negative square roots and infinity)

·      Integer Numbers  z { -3, -2, -1, 0, 1, 2, 3 }

·      Indicates what kind of number E

·      I – such that

Set builder provides the conditions that makes a function true  and provides a set of these conditions– A mathematical example would be:

F(x) = 1/8-x

We know that to ensure that the function will not be 0, we cannot have any numbers that would make the function zero: essentially 8 and lower

To represent this we would write:

D : { X e R I X  < 7 }  - this represents the input numbers for x – we use less than or equal to 7 to ensure that the result will always be positive.

R: { Y e R I Y > 0 }  - this represent the output as a result of the input numbers for x, they must be above 0 to make the equation work. 

The next way of writing domain and range is with interval notation.  Interval compared to set builder is much easier because it has a lot less rules for the function. Interval notation is much easier than set builder.   

Rules:

·      (    ) to write domain and range if you want to exclude the endpoints of the function – essentially only using < > and not including the number.

·      [    ]   including endpoints of the function – essentially using < >

·      You can also combine these two different types of brackets so for  [ 2, 8 ), the function includes two and goes to 8 but does not include 8 .

In mathematics following the same example as before:

F(x) = 1/ x - 8 

D: [7 , infinity ]

R:  [0, infinity]

The last and final example is roster notation which lists the element of the set inside brackets.

]

For this equation                          F(x) = 1/8-x

You just list all of the numbers that work for the function

{… -1, 0, 1, 2, 3, 4, 5, 6, 7,}

For further instruction, please visit this very helpful website:

https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-of-a-function