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Solve f(-x)=-f(x)?

  1. Aug 3, 2004 #1
    How do you use f(-x)=-f(x)???

    I understand how to use f(x)=f(-x)
    Ex:
    y=x^3
    =(-x)^3
    =(-x)(-x)(-x)
    =-x^3

    but Im confused with the previous.

    Any help and examples would be appreciated!!!

    trip7
     
  2. jcsd
  3. Aug 3, 2004 #2

    matt grime

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    what on earth do you mean by "use"?
     
  4. Aug 3, 2004 #3

    arildno

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    What are you talking about??
    What do you mean by "using"?
    The example used is wrong for all other choices than x=0.

    f(x)=f(-x) is the property of an even function (defined on some interval -a<=x<=a, a>0).

    f(-x)=-f(x) is the property of an odd function (defined on some interval -a<=x<=a, a>0).

    Usually, exercises involving even/odd require us to determine whether a given function is either even or odd (or neither or both).

    Let us take the case f(x)=x^3, and we assume it is defined on the whole number line.
    We will investigate whether f is even or odd:

    By the definition of f(x), we have:
    f(-x)=(-x)^3 (just plugging in -x at x's place in the expression for the function)
    We now manipulate the right-hand side:
    (-x)^3=(-1)^3*x^3=-x^3
    But, x^3=f(x), and we therefore have:
    f(-x)=-f(x)
    That is, f(x)=x^3 is is seen to be an odd function
     
  5. Aug 3, 2004 #4
    f(-x) = -f(x) = (-1)*f(x)
     
  6. Aug 3, 2004 #5
    Pardon my lack of proper terminology. Im trying to find even, odd, neither, or both. What I meant by 'using' is arildno's example "just plugging in -x at x's place in the expression for the function." Im trying to learn curve tracing and checking for symmetry I replaced x with -x aka f(x)=f(-x). I forgot how to find the origin and for some reason f(-x)=-f(x) comes to mind. Is there a simple technique like replace x with -x? Hope this isnt more confusing than previous post.

    trip7
     
  7. Aug 3, 2004 #6
    According to Gonzolo's equation, it appears that you just multiply all terms by -1. Is that correct? I really appreciate everyone help!!!

    trip7
     
  8. Aug 3, 2004 #7

    arildno

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    "Im trying to learn curve tracing and checking for symmetry I replaced x with -x aka f(x)=f(-x)."

    This is COMPLETELY, TOTALLY WRONG!!!!!
    Read what I wrote carefully:
    First, I EVALUATE f(-x)
    Then, I COMPARE the gained expression with what I know of f(x)
    Then I can draw a CONCLUSION if f is either odd, even or neither or both.

    Another example:
    Let g(x)=x^2
    Then, we EVALUATE: g(-x)=(-x)^2=(-1)^2*x^2=x^2
    But, COMPARING with g(x), I see that I can CONCLUDE:
    g(-x)=x^2=g(x), that is, g(-x)=g(x)
    That is, g is an even function.

    Yet another example:
    Let h(x)=x^2-x
    Then:
    h(-x)=(-x)^2-(-x)=x^2+x

    But, the expression x^2+x is not equal to h(x), nor is it equal to -h(x)!
    That is:
    h(x) is neither even or odd!
     
  9. Aug 3, 2004 #8
    Thanks for the extra examples arildno, I get it. I put way more into it than I should have!!!
     
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