1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to show a function is even/odd

  1. Aug 22, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi,
    I'm new to this site, I've had a look around and there are alot of useful sections, particularly the section with math and science learning materials.
    Anyway, I need to show that the following function is odd
    [tex]f(x)=\left\{\begin{array}{ccc}
    -\sin x&\mbox{ for }-\pi \leq x< \frac{-\pi}{ 2}\\
    \sin x &\mbox{ for } \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}\\
    -\sin x &\mbox{ for } \frac{\pi}{2}<x<\frac{\pi}{2}
    \end{array}\right.[/tex]

    [tex]\mbox{ and }f(x + 2 \pi) = f(x) \mbox{
    for all other values of x, is an odd function.}[/tex]


    2. Relevant equations

    I know an odd function is definded as [tex] f(-x) = -f(x)[/tex]

    3. The attempt at a solution
    In the interval
    [tex]-\pi\leq x < {-\pi \over 2} \mbox{ if I substiture } -\pi \mbox{ it becomes }-\sin(-x) = -\sin[-(-{\pi \over 2})] = -\sin({\pi \over 2})[/tex]

    Is that the correct way to solve it?
    But I'm not sure how to show it's odd in the other intervals!
     
    Last edited: Aug 22, 2009
  2. jcsd
  3. Aug 22, 2009 #2

    zcd

    User Avatar

    If f(-x)=f(x), then the function is even. If f(-x)=-f(x), then the function is odd.
     
  4. Aug 22, 2009 #3

    robphy

    User Avatar
    Science Advisor
    Homework Helper
    Insights Author
    Gold Member

    This might be a useful variation of the definitions:

    If the function f(x) is even, then f(x)-f(-x)=0 for all x.
    If the function f(x) is odd, then f(x)+f(-x)=0 for all x.
     
  5. Aug 22, 2009 #4
    To show it's odd:
    look at values in the intervals?
    [tex]-\sin(-\pi) - \sin({\pi }) = 0 [/tex]

    [tex] \sin({-\pi \over 2}) + \sin({\pi \over 2}) = 0[/tex]

    [tex]-\sin({3 \pi \over 4}) - \sin({-3 \pi \over 4}) = 0[/tex]

    do I need to show anything else?
     
  6. Aug 22, 2009 #5
    You would have to show that it's true for every value in the interval, not just at a few random points. So you'd have to let [tex]a[/tex] be a random value in each interval, and then look at [tex]f(a)[/tex] and [tex]f(-a)[/tex]. Since the intervals are symmetric, once you've assigned an interval for [tex]a[/tex], it will be obvious what interval [tex]-a[/tex] is in and therefore which definition of the function you need to use.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: How to show a function is even/odd
  1. Odd and even function (Replies: 5)

Loading...