To determine algebraically whether a function is odd, even, or neither, substitute (-x) for every x value in the function, simplify, and compare the result to the original function (f (x)).
If  f (-x) = f (x), the function is even. (results are the same)
If  f (-x) = -f (x), the function is odd. (results are negations)
If neither of these situations exists, the function is neither.

Function Comparisons Result
1.  f (x) = 3x2 + 1 f (-x) = 3(-x)2+1 = 3x2+1   (same)
f (-x) = f (x)		 EVEN
2.  f (x) = x3 - 2x f (-x) = (-x)3 - 2(-x) = -x3 + 2x 
  -x3 + 2x = -( x3 - 2x)  (negation of f (x))
f (-x) = -f (x)		 ODD
3.  f (x) = x2 - x f (-x) = (-x)2 -(-x) = x2 + x 
  x2 + x  ≠ x2 - x  (not even)   f (-x) ≠ f (x)
  x2 + x  ≠-(x2 - x) = -(x2) + x 
   (not odd)   f (-x) ≠ -f (x)
 NEITHER
4.
  f (-x) = f (x)
Since the exponents are even, the negative signs get gone.		 EVEN

5.


  f (-x) ≠ f (x)

  f (-x) ≠ -f (x)		 NEITHER
6.


(negation)   f (-x) = -f (x)

 ODD
7.  f (x) = | x - 6 | f (-x) = | (-x) - 6 | 
  | -x - 6 |  ≠ | x - 6 |   (not even)
  f (-x) ≠ f (x)
  | -x - 6 | ≠ - | x - 6 |   (not odd)
  f (-x) ≠ -f (x)		 NEITHER

 

hint gal
Remember: The trick to working with odd and even functions is to plug in (-x) in place of x in the original function, and see what happens.
If there is no change → even.   If a negation occurs → odd.

Note: In the world of functions, most functions are "neither".


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