Is the function $f(x)=(1/x^3+x)^5$ even, odd or neither?

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Answer 1 · 2015-11-17T08:06:49

$f(x) = (1/x^3 + x)^5$ is odd.

A function $f(x)$ is even if and only if $f(-x) = f(x)$ .
A function $f(x)$ is odd if and only if $f(-x) = -f(x)$ .

To check this function then, we will look at $f(-x)$ .

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

$=> f(-x) = ((-1)(1/x^3 + x))^5 = (-1)^5(1/x^3+x)^5$

$=> f(-x) = -(1/x^3 + x)^5 = -f(x)$

Thus in this case $f(x)$ is odd.

Is the function f(x)=(1/x^3+x)^5f(x)=(1/x^3+x)^5 even, odd or neither?