Test to determine if a function y=f(x) is even, odd or neither

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Test to determine if a function y=f(x) is even, odd or neither:Replace x with -x and compare the result to f(x). If f(-x) = f(x), the function is even. If f(-x) = - f(x), the function is odd. If f(-x) f(x) and f(-x) -f(x), the function is neither even nor odd. Remember that (!x)n=xn, if n is even!xn, if n is odd"#$%$So, taking (-x) to an evenpower gives: (-x)2 = (-x)(-x) = x2, (-x)4 = (-x)(-x)(-x)(-x) = x4, (-x)6= (-x)(-x)(-x)(-x)(-x)(-x) = x6 Taking (-x) to an oddpower gives: (-x)3=(-x)(-x)(-x) = - x3 , (-x)5= (-x)(-x)(-x)(-x)(-x) = - x5 , etc. Terms which involve oddpowers of x will change signswhen x is replaced with (-x). Terms which involve evenpowers of x willremain the samewhen x is replaced with (-x). And since constantterms do not involve x, they will also remain the samewhen x is replaced with (-x). __________________________________________________________________Examples:a) Functions whose terms contain only EVEN powers of the variable x and possibly a constant term (but no terms containing ODD powers of x) are likely to be EVEN functions. For example: f(x) = x4- 3x2+ 7 Replacing x with -x we obtain: f(-x) = (-x)4- 3(-x)2+ 7 = x4- 3x2+ 7 = f(x).

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