Even and Odd Functions of One Independent Variable

 

Criteria: 

 

     If  f(x) is odd then  f( ˗ x)  =  ˗ f(x)  and even if  f(˗ x )  = f(x).

 

 

Example:  Determine if the following function, f(x), is even or odd.

 

         f(x)  =  x3 +  sin(5x)

 

Replace  x  with  ˗ x.    Then  f(˗ x)  =  ( ˗ x)3 +  sin(˗ 5x)  =   ˗ x3 ˗  sin( 5x) 

 

 The result is that   f(x)  =  ˗ f(˗ x)

 

                So this function is odd.                        (result)

 

 

Example:  Determine if the following function, f(x), is even or odd.

 

        f(x)  =  x4 +  cos(5x)

 

Replace  x  with  ˗ x.      f(˗ x)  =  (˗ x)4 +  cos(˗ 5x)  =   x4 +  cos( 5x) 

 

Since  f(˗ x)  =   f(x)    the function is even.    (result)

 

 

Example:  Determine if the following function, f(x), is even or odd.

 

             f(x)  =  cos (x3)  ˗  sin(x4)     Replace  x  with  ˗ x.

 

 f(˗ x)  =  cos  ((˗ x)3)  ˗  sin ( (˗x)4 )  =  cos (˗ x3)  ˗  sin (x4)   = cos ( x3)  ˗  sin (x4) =  f(x)

 

Since  f(˗x)  =  f(x)  the function is even.       (result)

 

 

Example:  Determine if the following function, f(x), is even or odd.

 

             f(x)  =  cos (x3)  ˗  sin(x3)       Replace  x  with  ˗ x.

 

Here  f(˗ x)  =  cos ((˗ x)3)  ˗  sin( (˗ x)3)    =  cos (˗ x3)  ˗   sin (˗ x3)   =  cos (x3)  +   sin (x3)  

 

Thus:       f( ˗ x)  ≠  ˗ f(x)  and   f(˗ x )  ≠  f(x)

 

Therefore  f(x) is neither an even nor an odd function.                      (result)

 


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