0
$\begingroup$

For each of the following functions, state whether it is even or odd or periodic. If periodic, what is its smallest period?

  1. $\sin ax \quad (a>0)$
  2. $e^{ax} \quad (a>0)$
  3. $x^m \quad (m= \text{ integer})$
  4. $\tan x^2$
  5. $|\sin (x/b)| \quad (b>0)$
  6. $x\cos ax \quad (a>0)$
  • 0
    It makes sense. Can you answer the question? I have some ideas, but they will be lacking some nice detail that I want to make sure I have.2012-10-26

1 Answers 1

1

The definition of an even function is $f(-x) = f(x)$. The definition of an odd function is $f(-x) = -f(x)$. A periodic function means that for a fixed number $P$, $f(x + P) = f(x)$.

Therefore, substituting $-x$ in for $x$, $\sin(-ax) = -\sin(ax)$. This matches the definition of an odd function, so $\sin(ax)$ is odd.

To find if it is periodic, draw a graph of $\sin(ax)$ (I used WolframAlpha) and see if the graph repeats itself at all. In this case it does, every $2\pi$. Therefore the period is $2\pi$.

Try this method on the other functions and you should find it easy :)

  • 0
    @AsafKaragila : Thank you for the correction :)2012-10-26