1.1 Functions

According to the textbook, a function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs).

Characteristics of a function:
1. Each element in X must be matched with an element of Y.
2. Some elements in Y may not be matched with any element in X.
3. Two or more elements of X may be matched with the same element of Y.















This is a function because each x value corresponds to a single y value.                                                       

                                                      














This is not a function because for the value of 2, it corresponds with two different y values, B and C. 



Evaluating a function:
                                    
Ex.1      f(x) = 5x-14 Solve for f(2)        
             f(2)= 5(2)-14                          
             f(2)= 10-14
             f(2)= -4

Ex. 2    f(x) = x² + 2x - 7 Solve for f(-3)
            f(-3)= (-3)² + 2(-3) - 7
            f(-3)= 9 - 6 - 7
            f(-3)= -4



In order to determine whether a graph is a function, it must pass the vertical line test. A vertical line is placed on the graph, and in order for it to be a function, it must only intersect with the graph in one place across the entire graph, as shown in the first picture. The second picture is not a function because the line intersects at two different places.

A Piecewise-Defined Function

Ex. 1       Evaluate the function when x=3.







3 > 0 so...
f(3) = x - 2
f(3)= 3 - 2
f(3) =1