Notes For Calculus Life Saver

1.4 Odd and Even Functions

Some functions are symmetric, and it’ll be easy to discuss a function if it is symmetric. 

If all the x in its domain satisfy: f(x)=f(-x), we call function f an even function. In fact, for all functions like j(x)=x^n,n≡0(mod 2), j is always an even function. An even function, in fact, is symmetric about the y axis. It’s just like a mirror.

Similarly, if all the x in its domain satisfy: -f(x)=f(-x), then we call it an odd function. And for all the functions like: j(x)=x^n,n≡1(mod 2), they are always odd functions. An odd function, in fact, is symmetric about its point O(0,0). We can think that it rotates 180° around the point O.

There are odd, even functions. Some functions, however, is neither odd nor even. There is only one function that is both odd and even. We call it zero function.

There’s the proof:

Question: if function f is both odd and even, prove: f(x)=0

Prove: ∵f is odd

      ∴f(-(—x))=-f(-x)①

     ∵f is even

     ∴f(-(-x))=f(-x)②

①+②: 2f(-(—x))=0

                2f(x)=0

              f(x)=0

∴Quad Erat Demonstrandum