Relations and Functions



A relation is a rule that takes an input from a set (called the domain) and gives one or more outputs of another set (called the range).



  1.         (0,0), (4,4), (0,3), (2,1)

  2.         y = 2x

  3. The circle is a relation.  We can go from an x-coordinate to the y-coordinates of the circle that have that x-coordinate.



A function is a relation such that for every input there is exactly one output.



Examples 2 and 3 above are functions, whereas examples 1 and 4 are not functions.


Determining the Domain of a Function

Rule 1:  The domain of a polynomial is the set of all real numbers.




        f(x)  =  3x2 + 2x - 1

the domain is the set of all real numbers.


Rule 2:  The domain of a rational function (poly)/(poly) is the set of all real numbers except where the denominator is 0.




                          x - 1
        f(x)   =                                          
                          x + 1

the domain is all real numbers except where x  =  -1.


Rule 3:  The domain of a square root function is all real numbers that make the inside of the square root positive.


Find the domain of 




We set up  the inequality 

        x2 - x - 6  >  0 

and use our steps of quadratic inequalities to solve. Factoring we get

        (x - 3) (x + 2)  >  0

Putting -2 and 3 on a number line, gives three regions.  The table shows


x+2 x - 3 Total
Left (-3)












Hence the solution is 

        (-,-2]     U     [3,)


The Vertical Line Test

If every vertical line passes through the graph at most once then the graph is the graph of a function. 


The Circle is not the graph of a function since there are vertical lines that cross it twice.


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