How do you determine if ${(x - 1)}^{3} (x - 5)$ $(x-1)^3(x-5)$ is an even or odd function?

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Answer 1 · 2016-03-25T08:24:26

The given function is neither odd nor even.

If $f (- x) = f (x)$ $f(-x)=f(x)$ then the function is even,

but if $f (- x) = - f (x)$ $f(-x)=-f(x)$ then the function is odd.

As $f (x) = {(x - 1)}^{3} (x - 5)$ $f(x)=(x-1)^3(x-5)$

$f (- x) = {(- x - 1)}^{3} (- x - 5) = {(- (x + 1))}^{3} (- (x + 5))$ $f(-x)=(-x-1)^3(-x-5)=(-(x+1))^3(-(x+5))$ or

$f (- x) = (- {(x + 1)}^{3}) (- (x + 5)) = {(x + 1)}^{3} (x + 5)$ $f(-x)=(-(x+1)^3)(-(x+5))=(x+1)^3(x+5)$

Hence neither $f (- x) = f (x)$ $f(-x)=f(x)$ nor $f (- x) = - f (x)$ $f(-x)=-f(x)$

Hence the given function is neither odd nor even.

How do you determine if (x−1)3(x−5) (x-1)^3(x-5) is an even or odd function?