• Functions are the key to math!

Relation- A set of ordered pairs.

Function-  A relation where every input (x) has only one output (y). The X values do not repeat.

 Is it a function?

Solve For Y. 

+y=1 y=-+1 Yes                              

(This example is the first graph shown below.)

y=x  Yes      

(This example is the second graph shown below.)

-X+=1  y= No

(This example is last graph shown below.)

• The first two are clearly functions because of the definition of functions.  For example, if you put in 3 for X in the first example , then y= -8 , and nothing else. 
• The last example however is not a function.  If you put in -3 for X, then Y= 2 AND Y= -2, which contradicts the defintion of a function. 

Graphically speaking....

• Because there are not multiple Y values for each X values, no two points of the funciton will share an X coordinate.  This being said, if you draw a vertical line anywhere along your function, the line will NOT  cross the function more than once.  This is called the vertical line test. 
• http://www.teachertube.com/view_video.php?viewkey=f768bdf94162f550a2c3 is a TeacherTube video that illustrates the idea of this vertical line test.
• For more information about graphing functions, see section 2.2 Graphing Functions.

Function Notation

f(x)

f(X) is read as "F of X", and is another term for (y)

Any variable can be used in place of f, so you can have g(X) or m(X).

                                                                 example:                      h(X)=-+4x+1

Solve for  h(2) and h(X+2)

• To solve, plug in whatever is in the place of the X into the original equation.  

h(2)= -(2)^2+4(2)+1                                                          h(x+2)= -+4(x+2)+1

h(2)=    5                                                                        h(x+2)= --4x-4+4x+8+1

                                                                                      h(x+2)= -+5

Piecewise functions 

f(x)=  (X/2+1; X 1

          (3X+2; X  1  

• A piecewise function is actually two functions that begin or end only at a certain point.

 In this example, the first equation would be used IFF the X value that you were searching for was less than or equal to 1.  Otherwise, you will use the second equation. 

• A piecewise function will appear as two seperate functions on a graph, however there will only be one Y value for every X value.  The equation that does not include an "or equal to" sign will not have a filled in point where the two equations overlap.  

Domain

• The domain is simply the X values of your line or function.
• If you have the points (2,3), (5,6) and (11,20), then the domain is 2,5 and 11.
• The domain of any grap is simply all of the places where the line passes over an x value.

Domain Problems..

• If you have a fraction with an X in the denominator, then whatever number makes the X(the denominator) zero is not included in the domain.
• If you have a function with the square root of X, then the domain begins with number greater than zeros. 

Writing Domain

• Domains are wriiten in parenthesis, either (x,y) or [x,y].
• The curved parenthesis show that the number is not included in the domain, and the square parenthesis do the opposite.  For example, if you had (2,3], then the domain is anything Greater than two, and Less than or equal to three.
• When writing a domain, the lesser number is always first. 
• If there is a break between the domain, such as an exclusion caused by the domain problems above, make two seperate domains and link them with a "u" (union). 
• With a never ending function, use the infinity symbol with curved parentheis. (-,)

Examples!

Find the Domain

    (-,)                                                                                                  f(x)=   [0,)                   

f(x)= 1/(-1)      (where does the denominator = 0?)

Set the equation equal to zero, and solve for X.  Then, exclude these values from your domain.  In this example, you domain would be (-,-1)u(-1,1)u(1,).   As noted by the curved parenthesis, -1 and 1 are excluded from the domain.  

These two websites contain great help on this topic:

PurpleMath

Functions