3.1A   Even and Odd Functions

Even and Odd Functions from a Graph   

The words "even" and "odd," when applied to a function f(x), describe the symmetry that exists for the graph of the function.

 Even Functions

~  A function is even if and only if, whenever the point (x, y) is on the graph of f(x), 

then the point (-x, y) is also on the graph.

~  A function, f(x), is even if, for every number x in its domain, the number -x is also in the domain and f(-x) = f(x).

Notice: If a function is even, it is symmetric along the y-axis. Make sure you make this connection.

Ex 1:   f(x) = 2x4- x2 -1

       f(-x) = 2(-x)4 - (-x)2 - 1    =>   2x4 - x2 - 1  

       The result is the same as the original function. 

        This is an even function.

Ex 2:    f(x) = 2x4 - x

        f(-x) = 2(-x)4 - (-x)   =>  2x4 + x   

        Result is not the same as the original.

        This is not an even function.

~  A function is even if and only if its graph is symmetric with respect to the y-axis.