How do you determine whether $f (x) = (2 x^{5}) - (2 x^{3})$ $f(x)=(2x^5)-(2x^3)$ is an odd or even function?

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Answer 1 · 2018-05-22T20:33:59

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the definition of an odd function is the characteristic $f (- x) = - f (x)$ $f(-x) = -f(x)$ .

if you swap a certain $x$ $x$ -value for its additive inverse, then the $y$ $y$ -value will also change to its additive inverse.

here, $f (x) = 2 x^{5} - 2 x^{3}$ $f(x) = 2x^5 - 2x^3$ .

$f (- x)$ $f(-x)$ would be the new expression when all the $x$ $x$ s were swapped for $- x$ $-x$ s, so:

$f (- x) = 2 {(- x)}^{5} - 2 {(- x)}^{3}$ $f(-x) = 2(-x)^5 - 2(-x)^3$

${(- x)}^{5} = - x$ $(-x)^5 = -x$
${(- x)}^{3} = - x$ $(-x)^3 = -x$

hence, $f (- x) = - 2 x^{5} - (- 2 x^{3})$ $f(-x) = -2x^5 - (-2x^3)$ , or $- 2 x^{5} + 2 x^{3}$ $-2x^5 + 2x^3$ .

this is the negative of $f (x)$ $f(x)$ , which is $+ 2 x^{5} - 2 x^{3}$ $+2x^5 - 2x^3$ .

for the function $f (x) = 2 x^{5} - 2 x^{3}$ $f(x) = 2x^5 - 2x^3$ , $f (- x) = - f (x)$ $f(-x) = -f(x)$ .

therefore, $f (x) = 2 x^{5} - 2 x^{3}$ $f(x) = 2x^5 - 2x^3$ is an odd function.

How do you determine whether f(x)=(2x5)−(2x3)f(x)=(2x^5)-(2x^3) is an odd or even function?