How do you know if $f (x) = {(x^{2} + 8)}^{2}$ $f(x) = (x^2+8)^2$ is an even or odd function?

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How do you know if $f (x) = {(x^{2} + 8)}^{2}$ $f(x) = (x^2+8)^2$ is an even or odd function?

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Answer 1 · 2016-01-06T13:00:59

$f (x) = {(x^{2} + 8)}^{2}$ $f(x)=(x^2+8)^2$ is even and not odd

Explanation:

even functions
A function $f (x)$ $f(x)$ is even $\Leftrightarrow f (x) = f (- x) \forall x$ $hArr f(x)=f(-x) AAx$

For the specific case of $f (x) = {(x^{2} + 8)}^{2}$ $f(x)=(x^2+8)^2$
$XXX f (- x) = {({(- x)}^{2} + 8)}^{2}$ $color(white)("XXX")f(-x) = ((-x)^2+8)^2$
$XXXXXXX = {(x^{2} + 8)}^{2}$ $color(white)("XXXXXXX")=(x^2+8)^2$
$XXXXXXX = f (x)$ $color(white)("XXXXXXX")=f(x)$

So the function is even

$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX$ $bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX"))$

odd functions
A function $f (x)$ $f(x)$ is odd $\Leftrightarrow f (- x) = - f (x) \forall x$ $hArr f(-x)=-f(x) AAx$

For the specific case of $f (x) = {(x^{2} + 8)}^{2}$ $f(x)=(x^2+8)^2$
$XXX f (- x) (= f (x)) > 0 \forall x$ $color(white)("XXX")f(-x) (=f(x)) > 0 AAx$
Therefore
$XXX - f (x) < 0 \forall x$ $color(white)("XXX")-f(x) < 0 AAx$

$f (- x) \neq - f (x)$ $f(-x) != -f(x)$

So the function is not odd

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How do you know if $f (x) = {(x^{2} + 8)}^{2}$ $f(x) = (x^2+8)^2$ is an even or odd function?

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Explanation:

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