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Math Help - Show that m is an even number, then f is an even function.

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    Show 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) = xm.
    Show that if m is an even number, then f is an even function.
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    Re: Show that m is an even number, then f is an even function.

    If m is an even integer, then \displaystyle \begin{align*} (-x)^m = x^m \end{align*}. Isn't that the very definition of an even function? That \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 \begin{align*} x^{2n} \end{align*}, where n is some other integer.

    \displaystyle \begin{align*} (-x)^m &= (-x)^{2n} \\ &= \left[(-x)^2\right]^n \\ &= \left(x^2\right)^n \\ &= x^{2n} \\ &= x^m \end{align*}
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    Re: 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?
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    Re: Show that m is an even number, then f is an even function.

    Quote Originally Posted by Eraser147 View Post
    I am still a little bit confused here, how is the whole function itself proven to be positive though?
    You should know that if you square anything, it becomes nonnegative...
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    Re: Show that m is an even number, then f is an even function.

    Sorry, I mean even.
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    Re: Show that m is an even number, then f is an even function.

    Quote Originally Posted by Eraser147 View Post
    Sorry, I mean even.
    Because by writing the power as 2n, it's the same as squaring, then taking to the power of n. The squaring makes it positive.
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    Re: 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.
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    Re: Show that m is an even number, then f is an even function.

    Quote Originally Posted by Eraser147 View Post
    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.
    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 \begin{align*} f(-x) = f(x) \end{align*} for all x.
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    Re: 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.
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