My question is how can I show this? My problem states: Suppose m is an integer and f is the function defined byf(x) = x^{m. }

Show that if mis an even number, thenfis an even function.

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- October 1st 2012, 05:40 PMEraser147Show that m is an even number, then f is an even function.
My question is how can I show this? My problem states: Suppose m is an integer and f is the function defined by

**f(x) = x**^{m. }

Show that if *m*is an even number, then*f*is an even function. - October 1st 2012, 05:46 PMProve ItRe: Show that m is an even number, then f is an even function.
If m is an even integer, then . Isn't that the very definition of an even function? That ...

If you need a more rigorous proof, since m is an even integer, you can write it as , where n is some other integer.

- October 1st 2012, 06:13 PMEraser147Re: Show that m is an even number, then f is an even function.
I am still a little bit confused here, how is the whole function itself proven to be positive though?

- October 1st 2012, 06:17 PMProve ItRe: Show that m is an even number, then f is an even function.
- October 1st 2012, 06:18 PMEraser147Re: Show that m is an even number, then f is an even function.
Sorry, I mean even.

- October 1st 2012, 06:20 PMProve ItRe: Show that m is an even number, then f is an even function.
- October 1st 2012, 06:27 PMEraser147Re: Show that m is an even number, then f is an even function.
Huh? But that wouldn't justify whether the function equates to being even. I mean for example if we plug in some numbers such as 3 into the x. So it would look like 3^4*2. Yes, the n is even, BUT the function would equate to 6561. Sorry for my stupidity but please endure this with me.

- October 1st 2012, 07:03 PMProve ItRe: Show that m is an even number, then f is an even function.
An even function is NOT a function which only produces even numbers. An even function is a function that is a complete reflection of itself in the y - axis (in other words, inputting a negative number will give you the same answer as a positive one). In symbols, an even function has the property that for all x.

- October 1st 2012, 07:13 PMEraser147Re: Show that m is an even number, then f is an even function.
Makes a lot more sense. Thank you so much for your help.