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AlgebraII4You 1.1 (NOTES)

AlgebraII4You 1.1 (NOTES)

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Author: Bryce Schoenherr
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In this Algebra 2-4You tutorial, we will cover some different relations and functions.

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Tutorial

Relations and Functions

Concept to Know-Relations

   Don't worry.  Although this is in the title, it won't be a difficult section.  A relation is basically any group of ordered pairs.  For example, look at the points below.

open curly brackets open parentheses 1 comma 3 close parentheses semicolon open parentheses 2 comma 7 close parentheses semicolon open parentheses 4 comma 12 close parentheses semicolon open parentheses 2 comma 13 close parentheses close curly brackets


Concept to Know-Domain & Range

   In Algebra II, the list of 'x' numbers is called the domain (sometimes called the independent variables or inputs).  Likewise, the list of 'y' numbers is called the range (sometimes called dependent variables or outputs).

circle enclose open parentheses 1 close parentheses
open parentheses 2 close parentheses
open parentheses 4 close parentheses end enclose circle enclose open parentheses 3 close parentheses
open parentheses 7 close parentheses
open parentheses 12 close parentheses
open parentheses 13 close parentheses end enclose
D o m a i n R a n g e

   After you group the numbers together, you can connect them like this:



Concept to Know-Functions

   Functions are similar to relations, except that no two ordered pairs have the same first number.  That means that a number in the domain cannot point to two numbers in the range(therefore, the diagram above is NOT a function).  Another way to find if it is a function, works on a graph.  What you do is you take a vertical line and see if  it crosses more than one point on the line at once.  The one below is not a function.

Example

                                 WRONG



Concept to Know-Functional Notation

   Functional notation is represented by the equation f(x)=ax+b.  f(x) does not mean 'f' times 'x'.  It means the function of 'x'.  Sometimes this is interchangeable with y.  Let's say you have f(x)=2x+3, but the dreaded math textbook asks you to solve f(4)=2x+3.  All you do is substitute x for 4 to get an answer of 11.  Want to know what else you get out of that?  The ordered pair (4,11).


Hint-Functional Notation

   'f' may be substituted for 'g' or 'h'.  They may all be used if you have to graph multiple functions.  I bet you already know a form of this.  The area of a square is represented by this equation:

A left parenthesis r right parenthesis equals straight pi r squared

A=area, r=radius




Big Idea-Different Functions

Concept to Know-1 to 1 Functions

   RULE: Every 'y' value must have only one 'x' value.

   You may remember that we use the vertical line test to see if a line was a function.  To see if a line is a 1 to 1 function, we use the horizontal line test.  Although the line below is a function, it is not 1 to 1.

Example

                            WRONG


Concept to Know-Onto Functions

   RULE: Every 'y' value has an 'x' value.

   That means that if you use the horizontal line test, your line should hit at least one point on the graph at any point.

                           WRONG


Hint-"1 to 1" vs. "Onto"

   One way to remember them is:

   1-1: the horizontal line hits the graph at most once at any point

   Onto: the horizontal line hits the graph at least once at any point

Try It-"1 to 1" vs. "Onto"

   Let's first plot some different functions.

  • f(x)=x2
  • g(x)=2x
  • h(x)=x3-3x2-5x+6
  • j(x)=x3
f(x)=Appears to be neither "1 to 1" nor "onto"

g(x)=Appears to be neither "1 to 1" nor "onto"

h(x)Appears to be "onto", but not "1 to 1"

j(x)=Appears to be both "1 to 1" and "onto"

   We can also make a chart.  If there is "Yes" in 0 and/or 1, it's 1-1.  If it's 1 or more, it's onto.

Function Hits 0 spots Hits 1 spot Hits 2 spots Hits 3+ spots 1-1 Onto
f(x) Yes Yes Yes - - -
g(x) Yes Yes - Yes - -
h(x) - Yes Yes - - Yes
j(x) - Yes - - Yes Yes

   As you can see, a function can be neither or both too.


P.S. This bad drawing is why I copy and paste pictures usually.


Concept to Know-Discrete Functions

   xxx

Concept to Know-Continuous

   xxx



Summary

  • Relations
  • Domain and Range

  • Functions

  • Functional Notation

  • Linear Functions (COMING SOON)

Credits

Script Writer: Bryce Schoenherr

Head Mathematician: {PRIVATE}

Mathematicians: Bryce Schoenherr