Expert: Paul Klarreich - 10/11/2010

Question
determine algebraically whether each function is even, odd, or neither. f(x)=Cube root of 2x^2+1

Answer
Questioner:   Nick
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  finding if a function is even, odd, or neither
Question:  determine algebraically whether each function is even, odd, or neither. f(x)=Cube root of 2x^2+1
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1. f(x) is even if you write out f(-x) [BE SURE TO USE PARENTHESES.], do the algebra, and the answer is exactly the same as f(x).

2. f(x) is odd if you write out f(-x) [BE SURE TO USE PARENTHESES.], do the algebra, and the answer is exactly the opposite of f(x), meaning - f(x).

3. Most functions are neither.

4. Any function can be separated into an 'odd part' and an 'even  part'.  [Directions upon request.]

5. If a polynomial P(x) has only even powers of x, it is even.  [Zero is even.]

6. If a polynomial P(x) has only odd powers of x, it is odd.  [Zero is even.]

Now f(x) = (2x^2 + 1)^1/3  is even:

f(-x) = (2(-x)^2 + 1)^1/3

f(-x) = (2x^2 + 1)^1/3 = f(x)