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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- Oct 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. - Oct 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 $\displaystyle \displaystyle \begin{align*} (-x)^m = x^m \end{align*}$. Isn't that the very definition of an even function? That $\displaystyle \displaystyle \begin{align*} f(-x) = f(x) \end{align*}$...

If you need a more rigorous proof, since m is an even integer, you can write it as $\displaystyle \displaystyle \begin{align*} x^{2n} \end{align*}$, where n is some other integer.

$\displaystyle \displaystyle \begin{align*} (-x)^m &= (-x)^{2n} \\ &= \left[(-x)^2\right]^n \\ &= \left(x^2\right)^n \\ &= x^{2n} \\ &= x^m \end{align*}$ - Oct 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?

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

- Oct 1st 2012, 06:20 PMProve ItRe: Show that m is an even number, then f is an even function.
- Oct 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.

- Oct 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 $\displaystyle \displaystyle \begin{align*} f(-x) = f(x) \end{align*}$ for all x.

- Oct 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.