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I know that to determine wether or not a function is even, you sub in $(-x)$ for $x$, and see if it the same (even) or not (odd/neither). However, I get confused when you have to sub in $(-x)$ for multiple exponents of $x$ such as $x^9$ or higher.

I noticed that, as a general rule, if the exponent is even, such as $x^8$, the $-x$ will be $x$, and it will turn out to be $x^8$, however if the exponent is an odd number, such as $x^7$, the exponent will turn out to be $-x^7$, is this correct?

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    $\begingroup$ Yes, because $$(-1)^n = \left\{\begin{array}{lr} 1 &\text{$n$ even} \\ -1 &\text{$n$ odd}\end{array}\right.$$ $\endgroup$ – user61527 Jun 8 '14 at 6:50
  • $\begingroup$ Yes it is correct. You can prove your hypothesis by mathematical induction. $\endgroup$ – jdoicj Jun 8 '14 at 6:50
  • $\begingroup$ Yes, and it is related to why "even" and "odd" functions are called that way. $\endgroup$ – askyle Jun 8 '14 at 6:52
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Yes, because $$(-1)^n = \left\{\begin{array}{lr} 1 &\text{$n$ even} \\ -1 &\text{$n$ odd}\end{array}\right.$$

Combining this with the fact that

$$(-x)^n = ((-1) x)^n = (-1)^ n x^n$$

gives the desired result.

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