Question 129952
Remember, a function is even when this equality is true:


{{{f(x)=f(-x)}}}


and a function is odd when this equality is true:


{{{-f(x)=f(-x)}}}


Lets see if the function is even:


{{{f(x)=-3x^3+2x}}} Start with the given equation

Here's {{{f(x)}}}:


{{{f(x)=-3x^3+2x}}} This is the given equation


Here's {{{f(-x)}}}:


{{{f(-x)=-3(-x)^3+2(-x)}}} Replace x with {{{-x}}}


{{{f(-x)=3x^3-2x}}} Simplify


Since {{{-3x^3+2x}}} does not equal {{{3x^3-2x}}}, this means the equation {{{f(x)=f(-x)}}} is <b>not</b> true.


Since {{{f(x)}}} does <b>not</b> equal {{{f(-x)}}}, the equation {{{y=-3x^3+2x}}} is <b>not</b> an even function.



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Now lets see if the function is odd:


{{{y=-3x^3+2x}}} Start with the given equation


Here's {{{f(-x)}}}:


{{{f(-x)=3x^3-2x}}} Remember we solved for this previously


Here's {{{-f(x)}}}:


{{{-f(x)=-(-3x^3+2x)}}} Negate the whole function


{{{f(-x)=3x^3-2x}}} Distribute the negative and simplify

Since {{{3x^3-2x}}} equals {{{3x^3-2x}}}, this means the equation {{{-f(x)=f(-x)}}} is true.


Since {{{-f(x)}}} equals {{{f(-x)}}}, the equation {{{y=-3x^3+2x}}} is an odd function.


When we graph the equation {{{y=-3x^3+2x}}}, we can see if the equation has any symmetry:

{{{ graph( 500, 500, -10, 10, -10, 10, -3x^3+2x) }}}

and we can clearly see that the equation is <b>not</b>  symmetrical with respect to the y axis but is symmetrical with respect to the origin



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Answer:


So {{{f(x)=-3x^3+2x}}} is an odd function